%------------------------------------------------------------------------------
% File     : ITP215_3 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem Automation 00192_006252
% 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    : 0027_Automation_00192_006252 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 6684 (2534 unt;1214 typ;   0 def)
%            Number of atoms       : 11318 (5051 equ)
%            Maximal formula atoms :   17 (   2 avg)
%            Number of connectives : 6652 (1088   ~; 195   |; 509   &)
%                                         (1230 <=>;3630  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   4 avg)
%            Maximal term depth    :   37 (   2 avg)
%            Number of FOOLs       :  425 ( 246 fml; 179 var)
%            Number of X terms     :  234 (   0  []; 223 ite;  11 let)
%            Number of types       :  255 ( 254 usr)
%            Number of type conns  : 1237 ( 722   >; 515   *;   0   +;   0  <<)
%            Number of predicates  :  108 ( 105 usr;   2 prp; 0-6 aty)
%            Number of functors    :  862 ( 862 usr; 245 con; 0-5 aty)
%            Number of variables   : 12420 (12312   !; 108   ?;12420   :)
% 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
%            2022-02-17 15:59:57.857
%------------------------------------------------------------------------------
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tff(sy_c_List_Omeasures_001t__Int__Oint,type,
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tff(sy_c_List_Omeasures_001t__Nat__Onat,type,
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tff(sy_c_List_Omember_001t__Int__Oint,type,
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tff(sy_c_List_Omember_001t__Nat__Onat,type,
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tff(sy_c_List_On__lists_001t__Nat__Onat,type,
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tff(sy_c_List_Orotate1_001t__Assertions__Oassn,type,
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tff(sy_c_List_Orotate1_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
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tff(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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tff(sy_c_List_Osplice_001t__Int__Oint,type,
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tff(sy_c_List_Osplice_001t__Nat__Onat,type,
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tff(sy_c_List_Osubseqs_001t__Int__Oint,type,
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tff(sy_c_List_Osubseqs_001t__Nat__Onat,type,
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tff(sy_c_List_Otake_001t__Nat__Onat,type,
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tff(sy_c_List_Oupt,type,
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tff(sy_c_List_Oupto,type,
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tff(sy_c_List_Oupto__aux,type,
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tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001_Eo,type,
    aa_fun2484551872032894584nt_o_o: ( fun_fu967847188302805519nt_o_o * fun_Pr3238374608701176162_int_o ) > $o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
    aa_fun5349570537165229698_int_o: ( fun_fu3068557634114553409_int_o * fun_Pr3238374608701176162_int_o ) > fun_Pr3238374608701176162_int_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_M_Eo_J,type,
    aa_fun6202611221056298223nt_o_o: ( fun_fu7503463384463898374nt_o_o * fun_Pr3238374608701176162_int_o ) > fun_fu967847188302805519nt_o_o ).

tff(sy_c_aa_001t__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_001t__fun_It__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_Mt__fun_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
    aa_fun4287570227111185249_int_o: ( fun_fu959379683396315552_int_o * fun_Pr3238374608701176162_int_o ) > fun_fu3068557634114553409_int_o ).

tff(sy_c_fNot,type,
    fNot: fun_o_o ).

tff(sy_c_fequal_001_Eo,type,
    fequal_o: fun_o_fun_o_o ).

tff(sy_c_fequal_001t__Int__Oint,type,
    fequal_int: fun_int_fun_int_o ).

tff(sy_c_member_001t__Int__Oint,type,
    member_int2: ( int * set_int ) > $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__Nat__Onat,type,
    member_nat2: ( nat * set_nat ) > $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_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: ( product_prod_nat_nat * set_Pr1261947904930325089at_nat ) > $o ).

tff(sy_c_member_001t__fun_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    member7544390268018397336_nat_o: ( fun_Pr6460944699146269935_nat_o * set_fu3555266933746557007_nat_o ) > $o ).

tff(sy_v_F,type,
    f: assn ).

tff(sy_v_P,type,
    p: assn ).

tff(sy_v_Q,type,
    q: assn ).

% Relevant facts (5463)
tff(fact_0_assms,axiom,
    aa_assn_o(fi(nil_Pr5671120429643327159n_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),p),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),q),sln,sln),f) ).

% assms
tff(fact_1_FI__QUERY__def,axiom,
    ! [Pa: assn,Qa: assn,Fa: assn] :
      ( aa_assn_o(fI_QUERY(Pa,Qa),Fa)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Fa)) ) ).

% FI_QUERY_def
tff(fact_2_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ).

% assn_aci(10)
tff(fact_3_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) ).

% star_aci(3)
tff(fact_4_star__aci_I2_J,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).

% star_aci(2)
tff(fact_5_star__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% star_assoc
tff(fact_6_SLN__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,sln),Pa) = Pa ).

% SLN_left
tff(fact_7_SLN__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),sln) = Pa ).

% SLN_right
tff(fact_8_is__entails,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ).

% is_entails
tff(fact_9_ent__star__mono,axiom,
    ! [Pa: assn,P: assn,Qa: assn,Q: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),P)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Qa),Q)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P),Q)) ) ) ).

% ent_star_mono
tff(fact_10_ent__iffI,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),A2)
       => ( A2 = B2 ) ) ) ).

% ent_iffI
tff(fact_11_ent__refl,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Pa) ).

% ent_refl
tff(fact_12_ent__trans,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Qa),R)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R) ) ) ).

% ent_trans
tff(fact_13_assn__times__comm,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa) ).

% assn_times_comm
tff(fact_14_assn__times__assoc,axiom,
    ! [Pa: assn,Qa: assn,R: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),R) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),R)) ).

% assn_times_assoc
tff(fact_15_mult_Oright__assoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% mult.right_assoc
tff(fact_16_mult_Oright__assoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.right_assoc
tff(fact_17_mult_Oright__assoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.right_assoc
tff(fact_18_mult_Oright__assoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% mult.right_assoc
tff(fact_19_mult_Oright__assoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ).

% mult.right_assoc
tff(fact_20_mult_Oright__commute,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),B) ).

% mult.right_commute
tff(fact_21_mult_Oright__commute,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ).

% mult.right_commute
tff(fact_22_mult_Oright__commute,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ).

% mult.right_commute
tff(fact_23_mult_Oright__commute,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ).

% mult.right_commute
tff(fact_24_mult_Oright__commute,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),B) ).

% mult.right_commute
tff(fact_25_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% mult.assoc
tff(fact_26_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% mult.assoc
tff(fact_27_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% mult.assoc
tff(fact_28_mult_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% mult.assoc
tff(fact_29_mult_Oassoc,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ).

% mult.assoc
tff(fact_30_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_31_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_32_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_33_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_34_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_35_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_36_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_37_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_38_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_39_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_40_ent__wandI,axiom,
    ! [Qa: assn,Pa: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Pa)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),R)) ) ).

% ent_wandI
tff(fact_41_ent__mp,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa))),Qa) ).

% ent_mp
tff(fact_42_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_Pr3571187171380813162assn_o] : ~ aa_lis1404944545740551261assn_o(list_e7761433933450087034n_assn(Pa),nil_Pr5671120429643327159n_assn) ).

% list_ex1_simps(1)
tff(fact_43_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_int_o] : ~ aa_list_int_o(list_ex1_int(Pa),nil_int) ).

% list_ex1_simps(1)
tff(fact_44_list__ex1__simps_I1_J,axiom,
    ! [Pa: fun_nat_o] : ~ aa_list_nat_o(list_ex1_nat(Pa),nil_nat) ).

% list_ex1_simps(1)
tff(fact_45_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] : product_assn_assn(nil_assn,Uu) = nil_Pr5671120429643327159n_assn ).

% product.simps(1)
tff(fact_46_is__pure__assn__starI,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(is_pure_assn,A)
     => ( aa_assn_o(is_pure_assn,B)
       => aa_assn_o(is_pure_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ) ) ).

% is_pure_assn_starI
tff(fact_47_bind__simps_I1_J,axiom,
    ! [F: fun_Pr212062889907530505n_assn] : bind_P5664457578022171847n_assn(nil_Pr5671120429643327159n_assn,F) = nil_Pr5671120429643327159n_assn ).

% bind_simps(1)
tff(fact_48_bind__simps_I1_J,axiom,
    ! [F: fun_Pr4964005377303724544st_int] : bind_P8081679045764635286sn_int(nil_Pr5671120429643327159n_assn,F) = nil_int ).

% bind_simps(1)
tff(fact_49_bind__simps_I1_J,axiom,
    ! [F: fun_Pr7454088522235451300st_nat] : bind_P8084169516273685562sn_nat(nil_Pr5671120429643327159n_assn,F) = nil_nat ).

% bind_simps(1)
tff(fact_50_bind__simps_I1_J,axiom,
    ! [F: fun_in5179914838442496868n_assn] : bind_i4941608653131063484n_assn(nil_int,F) = nil_Pr5671120429643327159n_assn ).

% bind_simps(1)
tff(fact_51_bind__simps_I1_J,axiom,
    ! [F: fun_int_list_int] : bind_int_int(nil_int,F) = nil_int ).

% bind_simps(1)
tff(fact_52_bind__simps_I1_J,axiom,
    ! [F: fun_int_list_nat] : bind_int_nat(nil_int,F) = nil_nat ).

% bind_simps(1)
tff(fact_53_bind__simps_I1_J,axiom,
    ! [F: fun_na6580366267274133184n_assn] : bind_n3181571989377952280n_assn(nil_nat,F) = nil_Pr5671120429643327159n_assn ).

% bind_simps(1)
tff(fact_54_bind__simps_I1_J,axiom,
    ! [F: fun_nat_list_int] : bind_nat_int(nil_nat,F) = nil_int ).

% bind_simps(1)
tff(fact_55_bind__simps_I1_J,axiom,
    ! [F: fun_nat_list_nat] : bind_nat_nat(nil_nat,F) = nil_nat ).

% bind_simps(1)
tff(fact_56_member__rec_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn] : ~ aa_Pro2491721552408928739assn_o(member743271071679453132n_assn(nil_Pr5671120429643327159n_assn),Y) ).

% member_rec(2)
tff(fact_57_member__rec_I2_J,axiom,
    ! [Y: int] : ~ aa_int_o(member_int(nil_int),Y) ).

% member_rec(2)
tff(fact_58_member__rec_I2_J,axiom,
    ! [Y: nat] : ~ aa_nat_o(member_nat(nil_nat),Y) ).

% member_rec(2)
tff(fact_59_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) ) ) ).

% mult.safe_commute
tff(fact_60_mult_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,B),A) ) ) ).

% mult.safe_commute
tff(fact_61_mult_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(syntax380677951125133566nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) ) ) ).

% mult.safe_commute
tff(fact_62_mult_Osafe__commute,axiom,
    ! [X: code_natural,Y: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(syntax2868745204708187284atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Y)),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A) ) ) ).

% mult.safe_commute
tff(fact_63_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% mult.safe_commute
tff(fact_64_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y)),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A) ) ) ).

% assn_aci(11)
tff(fact_65_ent__pure__pre__iff,axiom,
    ! [Pa: assn,B: $o,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),pure_assn((B)))),Qa)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ) ).

% ent_pure_pre_iff
tff(fact_66_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_le1182329135810803268n_assn(N,nil_Pr5671120429643327159n_assn) = N ).

% gen_length_code(1)
tff(fact_67_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_int(N,nil_int) = N ).

% gen_length_code(1)
tff(fact_68_gen__length__code_I1_J,axiom,
    ! [N: nat] : gen_length_nat(N,nil_nat) = N ).

% gen_length_code(1)
tff(fact_69_maps__simps_I2_J,axiom,
    ! [F: fun_Pr212062889907530505n_assn] : maps_P4274076103917212289n_assn(F,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% maps_simps(2)
tff(fact_70_maps__simps_I2_J,axiom,
    ! [F: fun_Pr4964005377303724544st_int] : maps_P5983552678929597788sn_int(F,nil_Pr5671120429643327159n_assn) = nil_int ).

% maps_simps(2)
tff(fact_71_maps__simps_I2_J,axiom,
    ! [F: fun_Pr7454088522235451300st_nat] : maps_P5986043149438648064sn_nat(F,nil_Pr5671120429643327159n_assn) = nil_nat ).

% maps_simps(2)
tff(fact_72_maps__simps_I2_J,axiom,
    ! [F: fun_in5179914838442496868n_assn] : maps_i2843482286296025986n_assn(F,nil_int) = nil_Pr5671120429643327159n_assn ).

% maps_simps(2)
tff(fact_73_maps__simps_I2_J,axiom,
    ! [F: fun_int_list_int] : maps_int_int(F,nil_int) = nil_int ).

% maps_simps(2)
tff(fact_74_maps__simps_I2_J,axiom,
    ! [F: fun_int_list_nat] : maps_int_nat(F,nil_int) = nil_nat ).

% maps_simps(2)
tff(fact_75_maps__simps_I2_J,axiom,
    ! [F: fun_na6580366267274133184n_assn] : maps_n1083445622542914782n_assn(F,nil_nat) = nil_Pr5671120429643327159n_assn ).

% maps_simps(2)
tff(fact_76_maps__simps_I2_J,axiom,
    ! [F: fun_nat_list_int] : maps_nat_int(F,nil_nat) = nil_int ).

% maps_simps(2)
tff(fact_77_maps__simps_I2_J,axiom,
    ! [F: fun_nat_list_nat] : maps_nat_nat(F,nil_nat) = nil_nat ).

% maps_simps(2)
tff(fact_78_pure__assn__eq__conv,axiom,
    ! [Pa: $o,Qa: $o] :
      ( ( pure_assn((Pa)) = pure_assn((Qa)) )
    <=> ( (Pa)
      <=> (Qa) ) ) ).

% pure_assn_eq_conv
tff(fact_79_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,pure_assn((A))),pure_assn((B))) = pure_assn(( (A)
        & (B) )) ).

% merge_pure_star
tff(fact_80_is__pure__assn__pure,axiom,
    ! [Pa: $o] : aa_assn_o(is_pure_assn,pure_assn((Pa))) ).

% is_pure_assn_pure
tff(fact_81_is__pure__assn__def,axiom,
    ! [A: assn] :
      ( aa_assn_o(is_pure_assn,A)
    <=> ? [P2: $o] : A = pure_assn((P2)) ) ).

% is_pure_assn_def
tff(fact_82_is__pure__assnE,axiom,
    ! [A: assn] :
      ( aa_assn_o(is_pure_assn,A)
     => ~ ! [P3: $o] : A != pure_assn((P3)) ) ).

% is_pure_assnE
tff(fact_83_syntax__fo__nomatch__def,axiom,
    ! [Pat: assn,Obj: assn] : aa_assn_o(syntax7398250324933576852n_assn(Pat),Obj) ).

% syntax_fo_nomatch_def
tff(fact_84_ent__pure__post__iff,axiom,
    ! [Pa: assn,Qa: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),pure_assn((B))))
    <=> ( ! [H: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ) ).

% ent_pure_post_iff
tff(fact_85_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,pure_assn((B))),Qa)
    <=> ( (B)
       => aa_assn_o(aa_assn_fun_assn_o(entails,one_one_assn),Qa) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_86_ac__operator_Osafe__commute,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn,A: assn,B: assn] :
      ( syntax3673046151410588037r_assn(F)
     => ( aa_assn_o(syntax7398250324933576852n_assn(aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y)),A)
       => ( aa_assn_assn(aa_ass954072293579886259n_assn(F,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(F,B),A) ) ) ) ).

% ac_operator.safe_commute
tff(fact_87_and__extract__pure__left__ctx__iff,axiom,
    ! [Pa: assn,B: $o,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),pure_assn((B)))),Qa) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),pure_assn((B))) ).

% and_extract_pure_left_ctx_iff
tff(fact_88_and__extract__pure__right__ctx__iff,axiom,
    ! [Pa: assn,Qa: assn,B: $o] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),pure_assn((B)))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),pure_assn((B))) ).

% and_extract_pure_right_ctx_iff
tff(fact_89_null__rec_I2_J,axiom,
    aa_lis1404944545740551261assn_o(null_P7596217980778516671n_assn,nil_Pr5671120429643327159n_assn) ).

% null_rec(2)
tff(fact_90_null__rec_I2_J,axiom,
    aa_list_int_o(null_int,nil_int) ).

% null_rec(2)
tff(fact_91_null__rec_I2_J,axiom,
    aa_list_nat_o(null_nat,nil_nat) ).

% null_rec(2)
tff(fact_92_mem__Collect__eq,axiom,
    ! [A: int,Pa: fun_int_o] :
      ( member_int2(A,collect_int(Pa))
    <=> aa_int_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_93_mem__Collect__eq,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( member7544390268018397336_nat_o(A,collec6407654713642994138_nat_o(Pa))
    <=> aa_fun1675921005893230629at_o_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_94_mem__Collect__eq,axiom,
    ! [A: list_nat,Pa: fun_list_nat_o] :
      ( member_list_nat(A,collect_list_nat(Pa))
    <=> aa_list_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_95_mem__Collect__eq,axiom,
    ! [A: nat,Pa: fun_nat_o] :
      ( member_nat2(A,collect_nat(Pa))
    <=> aa_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_96_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,Pa: fun_Pr4796943543331071514_nat_o] :
      ( member8440522571783428010at_nat(A,collec3392354462482085612at_nat(Pa))
    <=> aa_Pro6150637604445964947_nat_o(Pa,A) ) ).

% mem_Collect_eq
tff(fact_97_Collect__mem__eq,axiom,
    ! [A2: set_int] : collect_int(aTP_Lamm_a(A2)) = A2 ).

% Collect_mem_eq
tff(fact_98_Collect__mem__eq,axiom,
    ! [A2: set_fu3555266933746557007_nat_o] : collec6407654713642994138_nat_o(aTP_Lamm_aa(A2)) = A2 ).

% Collect_mem_eq
tff(fact_99_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] : collect_list_nat(aTP_Lamm_ab(A2)) = A2 ).

% Collect_mem_eq
tff(fact_100_Collect__mem__eq,axiom,
    ! [A2: set_nat] : collect_nat(aTP_Lamm_ac(A2)) = A2 ).

% Collect_mem_eq
tff(fact_101_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : collec3392354462482085612at_nat(aTP_Lamm_ad(A2)) = A2 ).

% Collect_mem_eq
tff(fact_102_Collect__cong,axiom,
    ! [Pa: fun_fu5135969500543783852at_o_o,Qa: fun_fu5135969500543783852at_o_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( aa_fun1675921005893230629at_o_o(Pa,X2)
        <=> aa_fun1675921005893230629at_o_o(Qa,X2) )
     => ( collec6407654713642994138_nat_o(Pa) = collec6407654713642994138_nat_o(Qa) ) ) ).

% Collect_cong
tff(fact_103_Collect__cong,axiom,
    ! [Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ! [X2: list_nat] :
          ( aa_list_nat_o(Pa,X2)
        <=> aa_list_nat_o(Qa,X2) )
     => ( collect_list_nat(Pa) = collect_list_nat(Qa) ) ) ).

% Collect_cong
tff(fact_104_Collect__cong,axiom,
    ! [Pa: fun_nat_o,Qa: fun_nat_o] :
      ( ! [X2: nat] :
          ( aa_nat_o(Pa,X2)
        <=> aa_nat_o(Qa,X2) )
     => ( collect_nat(Pa) = collect_nat(Qa) ) ) ).

% Collect_cong
tff(fact_105_Collect__cong,axiom,
    ! [Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( ! [X2: product_prod_nat_nat] :
          ( aa_Pro6150637604445964947_nat_o(Pa,X2)
        <=> aa_Pro6150637604445964947_nat_o(Qa,X2) )
     => ( collec3392354462482085612at_nat(Pa) = collec3392354462482085612at_nat(Qa) ) ) ).

% Collect_cong
tff(fact_106_eq__Nil__null,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
    <=> aa_lis1404944545740551261assn_o(null_P7596217980778516671n_assn,Xs) ) ).

% eq_Nil_null
tff(fact_107_eq__Nil__null,axiom,
    ! [Xs: list_int] :
      ( ( Xs = nil_int )
    <=> aa_list_int_o(null_int,Xs) ) ).

% eq_Nil_null
tff(fact_108_eq__Nil__null,axiom,
    ! [Xs: list_nat] :
      ( ( Xs = nil_nat )
    <=> aa_list_nat_o(null_nat,Xs) ) ).

% eq_Nil_null
tff(fact_109_zip__Nil,axiom,
    ! [Ys: list_assn] : zip_assn_assn(nil_assn,Ys) = nil_Pr5671120429643327159n_assn ).

% zip_Nil
tff(fact_110_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr4327835960982145133n_assn = zip_Pr8806919808394484215n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_111_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( nil_Pr7991254345159696774sn_int = zip_Pr1015554993053626726sn_int(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_112_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( nil_Pr2945733327814117674sn_nat = zip_Pr1018045463562677002sn_nat(Xs,Ys) )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_113_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr2440414457004260852n_assn = zip_in7098856637274830732n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_114_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_Pr2300489316682597567nt_int = zip_int_int(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_115_Nil__eq__zip__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( nil_Pr6478340336191794275nt_nat = zip_int_nat(Xs,Ys) )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_116_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr680377793251149648n_assn = zip_na5338819973521719528n_assn(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_117_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( nil_Pr1301135604781543011at_int = zip_nat_int(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% Nil_eq_zip_iff
tff(fact_118_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat = zip_nat_nat(Xs,Ys) )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
tff(fact_119_Nil__eq__zip__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( nil_Pr5671120429643327159n_assn = zip_assn_assn(Xs,Ys) )
    <=> ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% Nil_eq_zip_iff
tff(fact_120_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_Pr8806919808394484215n_assn(Xs,Ys) = nil_Pr4327835960982145133n_assn )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_121_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( zip_Pr1015554993053626726sn_int(Xs,Ys) = nil_Pr7991254345159696774sn_int )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_122_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( zip_Pr1018045463562677002sn_nat(Xs,Ys) = nil_Pr2945733327814117674sn_nat )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_123_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_in7098856637274830732n_assn(Xs,Ys) = nil_Pr2440414457004260852n_assn )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_124_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( zip_int_int(Xs,Ys) = nil_Pr2300489316682597567nt_int )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_125_zip__eq__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( zip_int_nat(Xs,Ys) = nil_Pr6478340336191794275nt_nat )
    <=> ( ( Xs = nil_int )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_126_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( zip_na5338819973521719528n_assn(Xs,Ys) = nil_Pr680377793251149648n_assn )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_127_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( zip_nat_int(Xs,Ys) = nil_Pr1301135604781543011at_int )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_int ) ) ) ).

% zip_eq_Nil_iff
tff(fact_128_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( zip_nat_nat(Xs,Ys) = nil_Pr5478986624290739719at_nat )
    <=> ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
tff(fact_129_zip__eq__Nil__iff,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( zip_assn_assn(Xs,Ys) = nil_Pr5671120429643327159n_assn )
    <=> ( ( Xs = nil_assn )
        | ( Ys = nil_assn ) ) ) ).

% zip_eq_Nil_iff
tff(fact_130_mod__pure__star__dist,axiom,
    ! [Pa: assn,B: $o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),pure_assn((B)))),H2)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
        & (B) ) ) ).

% mod_pure_star_dist
tff(fact_131_list__ex__simps_I2_J,axiom,
    ! [Pa: fun_Pr3571187171380813162assn_o] : ~ aa_lis1404944545740551261assn_o(list_e1087220904399170805n_assn(Pa),nil_Pr5671120429643327159n_assn) ).

% list_ex_simps(2)
tff(fact_132_list__ex__simps_I2_J,axiom,
    ! [Pa: fun_int_o] : ~ aa_list_int_o(list_ex_int(Pa),nil_int) ).

% list_ex_simps(2)
tff(fact_133_list__ex__simps_I2_J,axiom,
    ! [Pa: fun_nat_o] : ~ aa_list_nat_o(list_ex_nat(Pa),nil_nat) ).

% list_ex_simps(2)
tff(fact_134_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_ass4243947775541419017_nat_o(rep_assn,X) = aa_ass4243947775541419017_nat_o(rep_assn,Y) )
    <=> ( X = Y ) ) ).

% Rep_assn_inject
tff(fact_135_mult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% mult_1
tff(fact_136_mult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% mult_1
tff(fact_137_mult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% mult_1
tff(fact_138_mult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% mult_1
tff(fact_139_mult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% mult_1
tff(fact_140_mult__1,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A) = A ).

% mult_1
tff(fact_141_mult_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.right_neutral
tff(fact_142_mult_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.right_neutral
tff(fact_143_mult_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.right_neutral
tff(fact_144_mult_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.right_neutral
tff(fact_145_mult_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.right_neutral
tff(fact_146_mult_Oright__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),one_one_Code_natural) = A ).

% mult.right_neutral
tff(fact_147_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,pure_assn((A))),pure_assn((B))) = pure_assn(( (A)
        & (B) )) ).

% merge_pure_and
tff(fact_148_pure__true,axiom,
    pure_assn($true) = one_one_assn ).

% pure_true
tff(fact_149_pure__assn__eq__emp__iff,axiom,
    ! [Pa: $o] :
      ( ( pure_assn((Pa)) = one_one_assn )
    <=> (Pa) ) ).

% pure_assn_eq_emp_iff
tff(fact_150_ent__pure__post__iff__sng,axiom,
    ! [Pa: assn,B: $o] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),pure_assn((B)))
    <=> ( ! [H: produc3658429121746597890et_nat] :
            ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
           => (B) )
        & aa_assn_o(aa_assn_fun_assn_o(entails,Pa),one_one_assn) ) ) ).

% ent_pure_post_iff_sng
tff(fact_151_and__extract__pure__right__iff,axiom,
    ! [Pa: assn,B: $o] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),pure_assn((B))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Pa)),pure_assn((B))) ).

% and_extract_pure_right_iff
tff(fact_152_and__extract__pure__left__iff,axiom,
    ! [B: $o,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,pure_assn((B))),Qa) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,one_one_assn),Qa)),pure_assn((B))) ).

% and_extract_pure_left_iff
tff(fact_153_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% norm_assertion_simps(14)
tff(fact_154_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% norm_assertion_simps(31)
tff(fact_155_assn__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% assn_aci(1)
tff(fact_156_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% assn_aci(3)
tff(fact_157_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% assn_aci(4)
tff(fact_158_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
    <=> ( X = one_one_Code_integer ) ) ).

% one_reorient
tff(fact_159_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
    <=> ( X = one_one_assn ) ) ).

% one_reorient
tff(fact_160_one__reorient,axiom,
    ! [X: rat] :
      ( ( one_one_rat = X )
    <=> ( X = one_one_rat ) ) ).

% one_reorient
tff(fact_161_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
    <=> ( X = one_one_nat ) ) ).

% one_reorient
tff(fact_162_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
    <=> ( X = one_one_int ) ) ).

% one_reorient
tff(fact_163_mod__and__dist,axiom,
    ! [Pa: assn,Qa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),H2)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H2) ) ) ).

% mod_and_dist
tff(fact_164_mod__starE,axiom,
    ! [A: assn,B: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),H2)
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A),X_1)
         => ! [H_2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B),H_2) ) ) ).

% mod_starE
tff(fact_165_mod__starD,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H2)
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),H1)
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),H22) ) ) ).

% mod_starD
tff(fact_166_entails__def,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
    <=> ! [H: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H)
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H) ) ) ).

% entails_def
tff(fact_167_entailsI,axiom,
    ! [Pa: assn,Qa: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H3)
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H3) )
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa) ) ).

% entailsI
tff(fact_168_entailsD,axiom,
    ! [Pa: assn,Qa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H2) ) ) ).

% entailsD
tff(fact_169_ent__fwd,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat,Qa: assn] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H2) ) ) ).

% ent_fwd
tff(fact_170_mult_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),one_one_rat) = A ).

% mult.comm_neutral
tff(fact_171_mult_Ocomm__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% mult.comm_neutral
tff(fact_172_mult_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),one_one_nat) = A ).

% mult.comm_neutral
tff(fact_173_mult_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),one_one_int) = A ).

% mult.comm_neutral
tff(fact_174_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),one_one_Code_integer) = A ).

% mult.comm_neutral
tff(fact_175_mult_Ocomm__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),one_one_Code_natural) = A ).

% mult.comm_neutral
tff(fact_176_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,one_one_rat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_177_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_178_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,one_one_nat),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_179_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,one_one_int),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_180_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,one_one_Code_integer),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_181_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,one_one_Code_natural),A) = A ).

% comm_monoid_mult_class.mult_1
tff(fact_182_ent__conjE2,axiom,
    ! [B2: assn,C2: assn,A2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C2) ) ).

% ent_conjE2
tff(fact_183_ent__conjE1,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B2)),C2) ) ).

% ent_conjE1
tff(fact_184_ent__conjI,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C2)
       => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B2),C2)) ) ) ).

% ent_conjI
tff(fact_185_assn__one__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),Pa) = Pa ).

% assn_one_left
tff(fact_186_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,one_one_assn),A) = A ).

% norm_assertion_simps(1)
tff(fact_187_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),one_one_assn) = A ).

% norm_assertion_simps(2)
tff(fact_188_is__pure__assn__basic__simps_I2_J,axiom,
    aa_assn_o(is_pure_assn,one_one_assn) ).

% is_pure_assn_basic_simps(2)
tff(fact_189_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
tff(fact_190_mult_Oac__operator__axioms,axiom,
    syntax3673046151410588037r_assn(times_times_assn) ).

% mult.ac_operator_axioms
tff(fact_191_mult_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat(times_times_nat) ).

% mult.ac_operator_axioms
tff(fact_192_mult_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int(times_times_int) ).

% mult.ac_operator_axioms
tff(fact_193_mult_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger(times_3573771949741848930nteger) ).

% mult.ac_operator_axioms
tff(fact_194_mult_Oac__operator__axioms,axiom,
    syntax8461159122685597829atural(times_2397367101498566445atural) ).

% mult.ac_operator_axioms
tff(fact_195_zip_Osimps_I1_J,axiom,
    ! [Xs: list_assn] : zip_assn_assn(Xs,nil_assn) = nil_Pr5671120429643327159n_assn ).

% zip.simps(1)
tff(fact_196_inf__apply,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,G: fun_Pr3238374608701176162_int_o,X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,F),G),X)
    <=> inf_inf_o(aa_Pro4592068669816069595_int_o(F,X),aa_Pro4592068669816069595_int_o(G,X)) ) ).

% inf_apply
tff(fact_197_inf__right__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_right_idem
tff(fact_198_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_right_idem
tff(fact_199_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_right_idem
tff(fact_200_inf__right__idem,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_right_idem
tff(fact_201_inf__right__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ).

% inf_right_idem
tff(fact_202_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.right_idem
tff(fact_203_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% inf.right_idem
tff(fact_204_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.right_idem
tff(fact_205_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.right_idem
tff(fact_206_inf_Oright__idem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ).

% inf.right_idem
tff(fact_207_inf__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_left_idem
tff(fact_208_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_left_idem
tff(fact_209_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_left_idem
tff(fact_210_inf__left__idem,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_left_idem
tff(fact_211_inf__left__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ).

% inf_left_idem
tff(fact_212_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ).

% inf.left_idem
tff(fact_213_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ).

% inf.left_idem
tff(fact_214_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ).

% inf.left_idem
tff(fact_215_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ).

% inf.left_idem
tff(fact_216_inf_Oleft__idem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ).

% inf.left_idem
tff(fact_217_inf__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),X) = X ).

% inf_idem
tff(fact_218_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),X) = X ).

% inf_idem
tff(fact_219_inf__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),X) = X ).

% inf_idem
tff(fact_220_inf__idem,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),X) = X ).

% inf_idem
tff(fact_221_inf__idem,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),X) = X ).

% inf_idem
tff(fact_222_inf_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),A) = A ).

% inf.idem
tff(fact_223_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),A) = A ).

% inf.idem
tff(fact_224_inf_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),A) = A ).

% inf.idem
tff(fact_225_inf_Oidem,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),A) = A ).

% inf.idem
tff(fact_226_inf_Oidem,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),A) = A ).

% inf.idem
tff(fact_227_inf__fun__def,axiom,
    ! [F: fun_Pr3238374608701176162_int_o,G: fun_Pr3238374608701176162_int_o,X3: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,F),G),X3)
    <=> inf_inf_o(aa_Pro4592068669816069595_int_o(F,X3),aa_Pro4592068669816069595_int_o(G,X3)) ) ).

% inf_fun_def
tff(fact_228_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) ).

% inf_sup_aci(4)
tff(fact_229_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_230_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_231_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) ).

% inf_sup_aci(4)
tff(fact_232_inf__sup__aci_I4_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) ).

% inf_sup_aci(4)
tff(fact_233_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_aci(3)
tff(fact_234_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_235_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_236_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_sup_aci(3)
tff(fact_237_inf__sup__aci_I3_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ).

% inf_sup_aci(3)
tff(fact_238_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_239_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_240_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_241_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_242_inf__sup__aci_I2_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ).

% inf_sup_aci(2)
tff(fact_243_inf__sup__aci_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_sup_aci(1)
tff(fact_244_inf__sup__aci_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_245_inf__sup__aci_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_246_inf__sup__aci_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ).

% inf_sup_aci(1)
tff(fact_247_inf__sup__aci_I1_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X) ).

% inf_sup_aci(1)
tff(fact_248_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).

% inf.assoc
tff(fact_249_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ).

% inf.assoc
tff(fact_250_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% inf.assoc
tff(fact_251_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ).

% inf.assoc
tff(fact_252_inf_Oassoc,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ).

% inf.assoc
tff(fact_253_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ).

% inf_assoc
tff(fact_254_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ).

% inf_assoc
tff(fact_255_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ).

% inf_assoc
tff(fact_256_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ).

% inf_assoc
tff(fact_257_inf__assoc,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ).

% inf_assoc
tff(fact_258_inf_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),A) ).

% inf.commute
tff(fact_259_inf_Ocommute,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),A) ).

% inf.commute
tff(fact_260_inf_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),A) ).

% inf.commute
tff(fact_261_inf_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),A) ).

% inf.commute
tff(fact_262_inf_Ocommute,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),A) ).

% inf.commute
tff(fact_263_inf__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X) ).

% inf_commute
tff(fact_264_inf__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X) ).

% inf_commute
tff(fact_265_inf__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X) ).

% inf_commute
tff(fact_266_inf__commute,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X) ).

% inf_commute
tff(fact_267_inf__commute,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X) ).

% inf_commute
tff(fact_268_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ).

% inf.left_commute
tff(fact_269_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),C)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ).

% inf.left_commute
tff(fact_270_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ).

% inf.left_commute
tff(fact_271_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ).

% inf.left_commute
tff(fact_272_inf_Oleft__commute,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),C)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ).

% inf.left_commute
tff(fact_273_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_left_commute
tff(fact_274_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_left_commute
tff(fact_275_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_left_commute
tff(fact_276_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_left_commute
tff(fact_277_inf__left__commute,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ).

% inf_left_commute
tff(fact_278_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: assn,K: assn,A: assn,B: assn] :
      ( ( A2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),A) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A2),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_279_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),A) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_280_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),A) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_281_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_282_boolean__algebra__cancel_Oinf1,axiom,
    ! [A2: fun_Pr3238374608701176162_int_o,K: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( A2 = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),A) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A2),B) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf1
tff(fact_283_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: assn,K: assn,B: assn,A: assn] :
      ( ( B2 = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),B) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B2) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,K),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_284_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B2 = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,K),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_285_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,K),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_286_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_287_boolean__algebra__cancel_Oinf2,axiom,
    ! [B2: fun_Pr3238374608701176162_int_o,K: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( ( B2 = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),B) )
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B2) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,K),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% boolean_algebra_cancel.inf2
tff(fact_288_mult_Omonoid__axioms,axiom,
    aa_rat_o(monoid_rat(times_times_rat),one_one_rat) ).

% mult.monoid_axioms
tff(fact_289_mult_Omonoid__axioms,axiom,
    aa_assn_o(monoid_assn(times_times_assn),one_one_assn) ).

% mult.monoid_axioms
tff(fact_290_mult_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(times_times_nat),one_one_nat) ).

% mult.monoid_axioms
tff(fact_291_mult_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(times_times_int),one_one_int) ).

% mult.monoid_axioms
tff(fact_292_mult_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(times_3573771949741848930nteger),one_one_Code_integer) ).

% mult.monoid_axioms
tff(fact_293_mult_Omonoid__axioms,axiom,
    aa_Code_natural_o(monoid_Code_natural(times_2397367101498566445atural),one_one_Code_natural) ).

% mult.monoid_axioms
tff(fact_294_ent__false__iff,axiom,
    ! [Pa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),bot_bot_assn)
    <=> ! [H: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H) ) ).

% ent_false_iff
tff(fact_295_mult_Ocomm__monoid__axioms,axiom,
    aa_rat_o(comm_monoid_rat(times_times_rat),one_one_rat) ).

% mult.comm_monoid_axioms
tff(fact_296_mult_Ocomm__monoid__axioms,axiom,
    aa_assn_o(comm_monoid_assn(times_times_assn),one_one_assn) ).

% mult.comm_monoid_axioms
tff(fact_297_mult_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(times_times_nat),one_one_nat) ).

% mult.comm_monoid_axioms
tff(fact_298_mult_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(times_times_int),one_one_int) ).

% mult.comm_monoid_axioms
tff(fact_299_mult_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(times_3573771949741848930nteger),one_one_Code_integer) ).

% mult.comm_monoid_axioms
tff(fact_300_mult_Ocomm__monoid__axioms,axiom,
    aa_Code_natural_o(comm_m478245049187884339atural(times_2397367101498566445atural),one_one_Code_natural) ).

% mult.comm_monoid_axioms
tff(fact_301_map__filter__simps_I2_J,axiom,
    ! [F: fun_Pr5046583358261358025n_assn] : map_fi2680429960852639882n_assn(F,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% map_filter_simps(2)
tff(fact_302_map__filter__simps_I2_J,axiom,
    ! [F: fun_Pr3977364135456993088on_int] : map_fi8666144670822621843sn_int(F,nil_Pr5671120429643327159n_assn) = nil_int ).

% map_filter_simps(2)
tff(fact_303_map__filter__simps_I2_J,axiom,
    ! [F: fun_Pr6467447280388719844on_nat] : map_fi8668635141331672119sn_nat(F,nil_Pr5671120429643327159n_assn) = nil_nat ).

% map_filter_simps(2)
tff(fact_304_map__filter__simps_I2_J,axiom,
    ! [F: fun_in2421953131120646948n_assn] : map_fi5526074278189050041n_assn(F,nil_int) = nil_Pr5671120429643327159n_assn ).

% map_filter_simps(2)
tff(fact_305_map__filter__simps_I2_J,axiom,
    ! [F: fun_int_option_int] : map_filter_int_int(F,nil_int) = nil_int ).

% map_filter_simps(2)
tff(fact_306_map__filter__simps_I2_J,axiom,
    ! [F: fun_int_option_nat] : map_filter_int_nat(F,nil_int) = nil_nat ).

% map_filter_simps(2)
tff(fact_307_map__filter__simps_I2_J,axiom,
    ! [F: fun_na5346875547889773184n_assn] : map_fi3766037614435938837n_assn(F,nil_nat) = nil_Pr5671120429643327159n_assn ).

% map_filter_simps(2)
tff(fact_308_map__filter__simps_I2_J,axiom,
    ! [F: fun_nat_option_int] : map_filter_nat_int(F,nil_nat) = nil_int ).

% map_filter_simps(2)
tff(fact_309_map__filter__simps_I2_J,axiom,
    ! [F: fun_nat_option_nat] : map_filter_nat_nat(F,nil_nat) = nil_nat ).

% map_filter_simps(2)
tff(fact_310_splice__Nil2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : splice4588524007259616650n_assn(Xs,nil_Pr5671120429643327159n_assn) = Xs ).

% splice_Nil2
tff(fact_311_splice__Nil2,axiom,
    ! [Xs: list_int] : splice_int(Xs,nil_int) = Xs ).

% splice_Nil2
tff(fact_312_splice__Nil2,axiom,
    ! [Xs: list_nat] : splice_nat(Xs,nil_nat) = Xs ).

% splice_Nil2
tff(fact_313_split__Nil__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( splice4588524007259616650n_assn(Xs,Ys) = nil_Pr5671120429643327159n_assn )
    <=> ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% split_Nil_iff
tff(fact_314_split__Nil__iff,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( splice_int(Xs,Ys) = nil_int )
    <=> ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% split_Nil_iff
tff(fact_315_split__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( splice_nat(Xs,Ys) = nil_nat )
    <=> ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% split_Nil_iff
tff(fact_316_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( rotate328796349445179396n_assn(Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
tff(fact_317_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( rotate1_int(Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
tff(fact_318_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( rotate1_nat(Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
tff(fact_319_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumer9200277086812776726n_assn(N,nil_Pr5671120429643327159n_assn) = nil_Pr680377793251149648n_assn ).

% enumerate_simps(1)
tff(fact_320_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumerate_int(N,nil_int) = nil_Pr1301135604781543011at_int ).

% enumerate_simps(1)
tff(fact_321_enumerate__simps_I1_J,axiom,
    ! [N: nat] : enumerate_nat(N,nil_nat) = nil_Pr5478986624290739719at_nat ).

% enumerate_simps(1)
tff(fact_322_mod__star__trueI,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
     => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H2) ) ).

% mod_star_trueI
tff(fact_323_mod__star__trueE,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H2)
     => ~ ! [H4: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H4) ) ).

% mod_star_trueE
tff(fact_324_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_right
tff(fact_325_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),bot_bo3516217578726220278_int_o) = bot_bo3516217578726220278_int_o ).

% boolean_algebra.conj_zero_right
tff(fact_326_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% boolean_algebra.conj_zero_right
tff(fact_327_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_right
tff(fact_328_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).

% boolean_algebra.conj_zero_right
tff(fact_329_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_zero_left
tff(fact_330_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,bot_bo3516217578726220278_int_o),X) = bot_bo3516217578726220278_int_o ).

% boolean_algebra.conj_zero_left
tff(fact_331_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% boolean_algebra.conj_zero_left
tff(fact_332_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% boolean_algebra.conj_zero_left
tff(fact_333_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).

% boolean_algebra.conj_zero_left
tff(fact_334_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% inf_bot_right
tff(fact_335_inf__bot__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),bot_bo3516217578726220278_int_o) = bot_bo3516217578726220278_int_o ).

% inf_bot_right
tff(fact_336_inf__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% inf_bot_right
tff(fact_337_inf__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),bot_bot_set_nat) = bot_bot_set_nat ).

% inf_bot_right
tff(fact_338_inf__bot__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),bot_bot_set_int) = bot_bot_set_int ).

% inf_bot_right
tff(fact_339_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),X) = bot_bo2099793752762293965at_nat ).

% inf_bot_left
tff(fact_340_inf__bot__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,bot_bo3516217578726220278_int_o),X) = bot_bo3516217578726220278_int_o ).

% inf_bot_left
tff(fact_341_inf__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% inf_bot_left
tff(fact_342_inf__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),X) = bot_bot_set_nat ).

% inf_bot_left
tff(fact_343_inf__bot__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),X) = bot_bot_set_int ).

% inf_bot_left
tff(fact_344_inf__top_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),top_to4669805908274784177at_nat) = A ).

% inf_top.right_neutral
tff(fact_345_inf__top_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),top_top_set_nat) = A ).

% inf_top.right_neutral
tff(fact_346_inf__top_Oright__neutral,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),top_to8487252020945707026_int_o) = A ).

% inf_top.right_neutral
tff(fact_347_inf__top_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),top_top_assn) = A ).

% inf_top.right_neutral
tff(fact_348_inf__top_Oright__neutral,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),top_to1996260823553986621t_unit) = A ).

% inf_top.right_neutral
tff(fact_349_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
    <=> ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_350_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
    <=> ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_351_inf__top_Oneutr__eq__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( top_to8487252020945707026_int_o = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
    <=> ( ( A = top_to8487252020945707026_int_o )
        & ( B = top_to8487252020945707026_int_o ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_352_inf__top_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( top_top_assn = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
    <=> ( ( A = top_top_assn )
        & ( B = top_top_assn ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_353_inf__top_Oneutr__eq__iff,axiom,
    ! [A: set_Product_unit,B: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),B) )
    <=> ( ( A = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.neutr_eq_iff
tff(fact_354_inf__top_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),A) = A ).

% inf_top.left_neutral
tff(fact_355_inf__top_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),A) = A ).

% inf_top.left_neutral
tff(fact_356_inf__top_Oleft__neutral,axiom,
    ! [A: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o),A) = A ).

% inf_top.left_neutral
tff(fact_357_inf__top_Oleft__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),A) = A ).

% inf_top.left_neutral
tff(fact_358_inf__top_Oleft__neutral,axiom,
    ! [A: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit),A) = A ).

% inf_top.left_neutral
tff(fact_359_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = top_to4669805908274784177at_nat )
    <=> ( ( A = top_to4669805908274784177at_nat )
        & ( B = top_to4669805908274784177at_nat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_360_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = top_top_set_nat )
    <=> ( ( A = top_top_set_nat )
        & ( B = top_top_set_nat ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_361_inf__top_Oeq__neutr__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = top_to8487252020945707026_int_o )
    <=> ( ( A = top_to8487252020945707026_int_o )
        & ( B = top_to8487252020945707026_int_o ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_362_inf__top_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = top_top_assn )
    <=> ( ( A = top_top_assn )
        & ( B = top_top_assn ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_363_inf__top_Oeq__neutr__iff,axiom,
    ! [A: set_Product_unit,B: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),B) = top_to1996260823553986621t_unit )
    <=> ( ( A = top_to1996260823553986621t_unit )
        & ( B = top_to1996260823553986621t_unit ) ) ) ).

% inf_top.eq_neutr_iff
tff(fact_364_top__eq__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( top_to4669805908274784177at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) )
    <=> ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% top_eq_inf_iff
tff(fact_365_top__eq__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( top_top_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) )
    <=> ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% top_eq_inf_iff
tff(fact_366_top__eq__inf__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( top_to8487252020945707026_int_o = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) )
    <=> ( ( X = top_to8487252020945707026_int_o )
        & ( Y = top_to8487252020945707026_int_o ) ) ) ).

% top_eq_inf_iff
tff(fact_367_top__eq__inf__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( top_top_assn = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) )
    <=> ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% top_eq_inf_iff
tff(fact_368_top__eq__inf__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( top_to1996260823553986621t_unit = aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) )
    <=> ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% top_eq_inf_iff
tff(fact_369_inf__eq__top__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = top_to4669805908274784177at_nat )
    <=> ( ( X = top_to4669805908274784177at_nat )
        & ( Y = top_to4669805908274784177at_nat ) ) ) ).

% inf_eq_top_iff
tff(fact_370_inf__eq__top__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = top_top_set_nat )
    <=> ( ( X = top_top_set_nat )
        & ( Y = top_top_set_nat ) ) ) ).

% inf_eq_top_iff
tff(fact_371_inf__eq__top__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = top_to8487252020945707026_int_o )
    <=> ( ( X = top_to8487252020945707026_int_o )
        & ( Y = top_to8487252020945707026_int_o ) ) ) ).

% inf_eq_top_iff
tff(fact_372_inf__eq__top__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = top_top_assn )
    <=> ( ( X = top_top_assn )
        & ( Y = top_top_assn ) ) ) ).

% inf_eq_top_iff
tff(fact_373_inf__eq__top__iff,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) = top_to1996260823553986621t_unit )
    <=> ( ( X = top_to1996260823553986621t_unit )
        & ( Y = top_to1996260823553986621t_unit ) ) ) ).

% inf_eq_top_iff
tff(fact_374_inf__top__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),top_to4669805908274784177at_nat) = X ).

% inf_top_right
tff(fact_375_inf__top__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ).

% inf_top_right
tff(fact_376_inf__top__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),top_to8487252020945707026_int_o) = X ).

% inf_top_right
tff(fact_377_inf__top__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).

% inf_top_right
tff(fact_378_inf__top__right,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),top_to1996260823553986621t_unit) = X ).

% inf_top_right
tff(fact_379_inf__top__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),X) = X ).

% inf_top_left
tff(fact_380_inf__top__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),X) = X ).

% inf_top_left
tff(fact_381_inf__top__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o),X) = X ).

% inf_top_left
tff(fact_382_inf__top__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X) = X ).

% inf_top_left
tff(fact_383_inf__top__left,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit),X) = X ).

% inf_top_left
tff(fact_384_merge__true__star,axiom,
    aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),top_top_assn) = top_top_assn ).

% merge_true_star
tff(fact_385_star__false__right,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),bot_bot_assn) = bot_bot_assn ).

% star_false_right
tff(fact_386_star__false__left,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,bot_bot_assn),Pa) = bot_bot_assn ).

% star_false_left
tff(fact_387_pure__false,axiom,
    pure_assn($false) = bot_bot_assn ).

% pure_false
tff(fact_388_pure__assn__eq__false__iff,axiom,
    ! [Pa: $o] :
      ( ( pure_assn((Pa)) = bot_bot_assn )
    <=> ~ (Pa) ) ).

% pure_assn_eq_false_iff
tff(fact_389_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
tff(fact_390_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
tff(fact_391_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
tff(fact_392_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m5709050211238513046at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat) ).

% inf_top.comm_monoid_axioms
tff(fact_393_inf__top_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat(inf_inf_set_nat,top_top_set_nat) ).

% inf_top.comm_monoid_axioms
tff(fact_394_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m285090526141560045_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.comm_monoid_axioms
tff(fact_395_inf__top_Ocomm__monoid__axioms,axiom,
    aa_assn_o(comm_monoid_assn(inf_inf_assn),top_top_assn) ).

% inf_top.comm_monoid_axioms
tff(fact_396_inf__top_Ocomm__monoid__axioms,axiom,
    comm_m7591727228438362018t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).

% inf_top.comm_monoid_axioms
tff(fact_397_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),top_to4669805908274784177at_nat) = X ).

% boolean_algebra.conj_one_right
tff(fact_398_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),top_top_set_nat) = X ).

% boolean_algebra.conj_one_right
tff(fact_399_boolean__algebra_Oconj__one__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),top_to8487252020945707026_int_o) = X ).

% boolean_algebra.conj_one_right
tff(fact_400_boolean__algebra_Oconj__one__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).

% boolean_algebra.conj_one_right
tff(fact_401_boolean__algebra_Oconj__one__right,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),top_to1996260823553986621t_unit) = X ).

% boolean_algebra.conj_one_right
tff(fact_402_inf__top_Omonoid__axioms,axiom,
    monoid1519094961268053602at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat) ).

% inf_top.monoid_axioms
tff(fact_403_inf__top_Omonoid__axioms,axiom,
    monoid_set_nat(inf_inf_set_nat,top_top_set_nat) ).

% inf_top.monoid_axioms
tff(fact_404_inf__top_Omonoid__axioms,axiom,
    monoid8055414245428884769_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.monoid_axioms
tff(fact_405_inf__top_Omonoid__axioms,axiom,
    aa_assn_o(monoid_assn(inf_inf_assn),top_top_assn) ).

% inf_top.monoid_axioms
tff(fact_406_inf__top_Omonoid__axioms,axiom,
    monoid202360481972076142t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).

% inf_top.monoid_axioms
tff(fact_407_merge__true__star__ctx,axiom,
    ! [Pa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),Pa)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,top_top_assn),Pa) ).

% merge_true_star_ctx
tff(fact_408_mod__false,axiom,
    ! [H2: produc3658429121746597890et_nat] : ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,bot_bot_assn),H2) ).

% mod_false
tff(fact_409_ent__false,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,bot_bot_assn),Pa) ).

% ent_false
tff(fact_410_ent__true,axiom,
    ! [Pa: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,Pa),top_top_assn) ).

% ent_true
tff(fact_411_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,bot_bot_assn),X) = bot_bot_assn ).

% norm_assertion_simps(9)
tff(fact_412_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),bot_bot_assn) = bot_bot_assn ).

% norm_assertion_simps(10)
tff(fact_413_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,top_top_assn),X) = X ).

% norm_assertion_simps(3)
tff(fact_414_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),top_top_assn) = X ).

% norm_assertion_simps(4)
tff(fact_415_is__pure__assn__basic__simps_I1_J,axiom,
    aa_assn_o(is_pure_assn,bot_bot_assn) ).

% is_pure_assn_basic_simps(1)
tff(fact_416_rotate1_Osimps_I1_J,axiom,
    rotate328796349445179396n_assn(nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% rotate1.simps(1)
tff(fact_417_rotate1_Osimps_I1_J,axiom,
    rotate1_int(nil_int) = nil_int ).

% rotate1.simps(1)
tff(fact_418_rotate1_Osimps_I1_J,axiom,
    rotate1_nat(nil_nat) = nil_nat ).

% rotate1.simps(1)
tff(fact_419_splice_Osimps_I1_J,axiom,
    ! [Ys: list_P8527749157015355191n_assn] : splice4588524007259616650n_assn(nil_Pr5671120429643327159n_assn,Ys) = Ys ).

% splice.simps(1)
tff(fact_420_splice_Osimps_I1_J,axiom,
    ! [Ys: list_int] : splice_int(nil_int,Ys) = Ys ).

% splice.simps(1)
tff(fact_421_splice_Osimps_I1_J,axiom,
    ! [Ys: list_nat] : splice_nat(nil_nat,Ys) = Ys ).

% splice.simps(1)
tff(fact_422_entt__def__true,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(entailst(Pa),Qa)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),top_top_assn)) ) ).

% entt_def_true
tff(fact_423_entailst__def,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),B2)
    <=> aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn)) ) ).

% entailst_def
tff(fact_424_enttI__true,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),top_top_assn))
     => aa_assn_o(entailst(Pa),Qa) ) ).

% enttI_true
tff(fact_425_enttI,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn))
     => aa_assn_o(entailst(A2),B2) ) ).

% enttI
tff(fact_426_enttD,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn)) ) ).

% enttD
tff(fact_427_entt__star__true__simp_I2_J,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),top_top_assn)),B2)
    <=> aa_assn_o(entailst(A2),B2) ) ).

% entt_star_true_simp(2)
tff(fact_428_entt__star__true__simp_I1_J,axiom,
    ! [A2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),top_top_assn))
    <=> aa_assn_o(entailst(A2),B2) ) ).

% entt_star_true_simp(1)
tff(fact_429_inf__top_Osemilattice__neutr__axioms,axiom,
    semila858173770281612099at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_430_inf__top_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat(inf_inf_set_nat,top_top_set_nat) ).

% inf_top.semilattice_neutr_axioms
tff(fact_431_inf__top_Osemilattice__neutr__axioms,axiom,
    semila8824639500271816704_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o) ).

% inf_top.semilattice_neutr_axioms
tff(fact_432_inf__top_Osemilattice__neutr__axioms,axiom,
    aa_assn_o(semila5388846958306562758r_assn(inf_inf_assn),top_top_assn) ).

% inf_top.semilattice_neutr_axioms
tff(fact_433_inf__top_Osemilattice__neutr__axioms,axiom,
    semila7638931295954862799t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit) ).

% inf_top.semilattice_neutr_axioms
tff(fact_434_entt__refl,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),A2) ).

% entt_refl
tff(fact_435_entt__true,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),top_top_assn) ).

% entt_true
tff(fact_436_entt__emp,axiom,
    ! [A2: assn] : aa_assn_o(entailst(A2),one_one_assn) ).

% entt_emp
tff(fact_437_entt__trans,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => ( aa_assn_o(entailst(B2),C2)
       => aa_assn_o(entailst(A2),C2) ) ) ).

% entt_trans
tff(fact_438_entt__frame__fwd,axiom,
    ! [Pa: assn,Qa: assn,A2: assn,Fa: assn,B2: assn] :
      ( aa_assn_o(entailst(Pa),Qa)
     => ( aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Fa))
       => ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Qa),Fa)),B2)
         => aa_assn_o(entailst(A2),B2) ) ) ) ).

% entt_frame_fwd
tff(fact_439_entt__star__mono,axiom,
    ! [A2: assn,B2: assn,C2: assn,D: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => ( aa_assn_o(entailst(C2),D)
       => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),D)) ) ) ).

% entt_star_mono
tff(fact_440_ent__imp__entt,axiom,
    ! [Pa: assn,Qa: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,Pa),Qa)
     => aa_assn_o(entailst(Pa),Qa) ) ).

% ent_imp_entt
tff(fact_441_mod__true,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),H2)
    <=> aa_Pro653082453696054616_nat_o(in_range,H2) ) ).

% mod_true
tff(fact_442_mod__h__bot__iff_I6_J,axiom,
    ! [Pa: assn,Qa: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(6)
tff(fact_443_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,pure_assn((B))),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> (B) ) ).

% mod_h_bot_iff(1)
tff(fact_444_in__range__empty,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% in_range_empty
tff(fact_445_mod__h__bot__iff_I5_J,axiom,
    ! [Pa: assn,Qa: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(5)
tff(fact_446_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7507926704131184380et_nat(H3,As) ).

% one_assn_raw.cases
tff(fact_447_mod__h__bot__indep,axiom,
    ! [Pa: assn,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H5,bot_bot_set_nat)) ) ).

% mod_h_bot_indep
tff(fact_448_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn(times_times_assn) ).

% mult.abel_semigroup_axioms
tff(fact_449_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat(times_times_nat) ).

% mult.abel_semigroup_axioms
tff(fact_450_mult_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int(times_times_int) ).

% mult.abel_semigroup_axioms
tff(fact_451_mult_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger(times_3573771949741848930nteger) ).

% mult.abel_semigroup_axioms
tff(fact_452_mult_Oabel__semigroup__axioms,axiom,
    abel_s7595615374776563986atural(times_2397367101498566445atural) ).

% mult.abel_semigroup_axioms
tff(fact_453_models__in__range,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
     => aa_Pro653082453696054616_nat_o(in_range,H2) ) ).

% models_in_range
tff(fact_454_inf_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn(inf_inf_assn) ).

% inf.abel_semigroup_axioms
tff(fact_455_inf_Oabel__semigroup__axioms,axiom,
    abel_s1357164767337380599at_nat(inf_in2572325071724192079at_nat) ).

% inf.abel_semigroup_axioms
tff(fact_456_inf_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat(inf_inf_set_nat) ).

% inf.abel_semigroup_axioms
tff(fact_457_inf_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat(inf_inf_nat) ).

% inf.abel_semigroup_axioms
tff(fact_458_inf_Oabel__semigroup__axioms,axiom,
    abel_s1734634174894267212_int_o(inf_in1782598092406300404_int_o) ).

% inf.abel_semigroup_axioms
tff(fact_459_mult_Osemigroup__axioms,axiom,
    semigroup_assn(times_times_assn) ).

% mult.semigroup_axioms
tff(fact_460_mult_Osemigroup__axioms,axiom,
    semigroup_nat(times_times_nat) ).

% mult.semigroup_axioms
tff(fact_461_mult_Osemigroup__axioms,axiom,
    semigroup_int(times_times_int) ).

% mult.semigroup_axioms
tff(fact_462_mult_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger(times_3573771949741848930nteger) ).

% mult.semigroup_axioms
tff(fact_463_mult_Osemigroup__axioms,axiom,
    semigr5859692871497579482atural(times_2397367101498566445atural) ).

% mult.semigroup_axioms
tff(fact_464_inf_Osemilattice__axioms,axiom,
    semilattice_assn(inf_inf_assn) ).

% inf.semilattice_axioms
tff(fact_465_inf_Osemilattice__axioms,axiom,
    semila6306352305301525377at_nat(inf_in2572325071724192079at_nat) ).

% inf.semilattice_axioms
tff(fact_466_inf_Osemilattice__axioms,axiom,
    semilattice_set_nat(inf_inf_set_nat) ).

% inf.semilattice_axioms
tff(fact_467_inf_Osemilattice__axioms,axiom,
    semilattice_nat(inf_inf_nat) ).

% inf.semilattice_axioms
tff(fact_468_inf_Osemilattice__axioms,axiom,
    semila6949790524937994306_int_o(inf_in1782598092406300404_int_o) ).

% inf.semilattice_axioms
tff(fact_469_inf_Osemigroup__axioms,axiom,
    semigroup_assn(inf_inf_assn) ).

% inf.semigroup_axioms
tff(fact_470_inf_Osemigroup__axioms,axiom,
    semigr3814207799437385263at_nat(inf_in2572325071724192079at_nat) ).

% inf.semigroup_axioms
tff(fact_471_inf_Osemigroup__axioms,axiom,
    semigroup_set_nat(inf_inf_set_nat) ).

% inf.semigroup_axioms
tff(fact_472_inf_Osemigroup__axioms,axiom,
    semigroup_nat(inf_inf_nat) ).

% inf.semigroup_axioms
tff(fact_473_inf_Osemigroup__axioms,axiom,
    semigr4744264435902949908_int_o(inf_in1782598092406300404_int_o) ).

% inf.semigroup_axioms
tff(fact_474_mod__h__bot__iff_I2_J,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,top_top_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% mod_h_bot_iff(2)
tff(fact_475_mod__emp__simp,axiom,
    ! [H2: heap_e7401611519738050253t_unit] : aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ).

% mod_emp_simp
tff(fact_476_pure__assn__raw_Oelims_I3_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H3,As) )
           => ( ( As = bot_bot_set_nat )
              & (X) ) ) ) ).

% pure_assn_raw.elims(3)
tff(fact_477_pure__assn__raw_Oelims_I2_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H3,As) )
           => ~ ( ( As = bot_bot_set_nat )
                & (X) ) ) ) ).

% pure_assn_raw.elims(2)
tff(fact_478_pure__assn__raw_Oelims_I1_J,axiom,
    ! [X: $o,Xa: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((X)),Xa)
      <=> (Y) )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xa = produc7507926704131184380et_nat(H3,As) )
           => ( (Y)
            <=> ~ ( ( As = bot_bot_set_nat )
                  & (X) ) ) ) ) ).

% pure_assn_raw.elims(1)
tff(fact_479_pure__assn__raw_Osimps,axiom,
    ! [B: $o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(pure_a825153325127701367it_nat((B)),produc7507926704131184380et_nat(H2,As2))
    <=> ( ( As2 = bot_bot_set_nat )
        & (B) ) ) ).

% pure_assn_raw.simps
tff(fact_480_mod__star__conv,axiom,
    ! [A2: assn,B2: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),B2)),H2)
    <=> ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
          ( ( H2 = produc7507926704131184380et_nat(Hr,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22)) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,A2),produc7507926704131184380et_nat(Hr,As1))
          & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,B2),produc7507926704131184380et_nat(Hr,As22)) ) ) ).

% mod_star_conv
tff(fact_481_star__assnI,axiom,
    ! [Pa: assn,H2: heap_e7401611519738050253t_unit,As2: set_nat,Qa: assn,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,As2))
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),produc7507926704131184380et_nat(H2,As3))
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As3) = bot_bot_set_nat )
         => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa)),produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3))) ) ) ) ).

% star_assnI
tff(fact_482_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
tff(fact_483_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.right_idem
tff(fact_484_sup_Oright__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.right_idem
tff(fact_485_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% sup_left_idem
tff(fact_486_sup__left__idem,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% sup_left_idem
tff(fact_487_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ).

% sup.left_idem
tff(fact_488_sup_Oleft__idem,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ).

% sup.left_idem
tff(fact_489_sup__idem,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),X) = X ).

% sup_idem
tff(fact_490_sup__idem,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% sup_idem
tff(fact_491_sup_Oidem,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),A) = A ).

% sup.idem
tff(fact_492_sup_Oidem,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),A) = A ).

% sup.idem
tff(fact_493_sup__bot_Oright__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),bot_bot_assn) = A ).

% sup_bot.right_neutral
tff(fact_494_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),bot_bot_set_nat) = A ).

% sup_bot.right_neutral
tff(fact_495_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),bot_bot_set_int) = A ).

% sup_bot.right_neutral
tff(fact_496_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_497_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_498_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
tff(fact_499_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),A) = A ).

% sup_bot.left_neutral
tff(fact_500_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),A) = A ).

% sup_bot.left_neutral
tff(fact_501_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),A) = A ).

% sup_bot.left_neutral
tff(fact_502_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = bot_bot_assn )
    <=> ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_503_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = bot_bot_set_nat )
    <=> ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_504_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),B) = bot_bot_set_int )
    <=> ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
tff(fact_505_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = bot_bot_assn )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
tff(fact_506_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = bot_bot_set_nat )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
tff(fact_507_sup__eq__bot__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = bot_bot_set_int )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
tff(fact_508_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) )
    <=> ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
tff(fact_509_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) )
    <=> ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
tff(fact_510_bot__eq__sup__iff,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) )
    <=> ( ( X = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
tff(fact_511_sup__bot__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% sup_bot_right
tff(fact_512_sup__bot__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),bot_bot_set_nat) = X ).

% sup_bot_right
tff(fact_513_sup__bot__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),bot_bot_set_int) = X ).

% sup_bot_right
tff(fact_514_sup__bot__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% sup_bot_left
tff(fact_515_sup__bot__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,bot_bot_set_nat),X) = X ).

% sup_bot_left
tff(fact_516_sup__bot__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,bot_bot_set_int),X) = X ).

% sup_bot_left
tff(fact_517_sup__top__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),top_top_set_nat) = top_top_set_nat ).

% sup_top_right
tff(fact_518_sup__top__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).

% sup_top_right
tff(fact_519_sup__top__right,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),top_to1996260823553986621t_unit) = top_to1996260823553986621t_unit ).

% sup_top_right
tff(fact_520_sup__top__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,top_top_set_nat),X) = top_top_set_nat ).

% sup_top_left
tff(fact_521_sup__top__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).

% sup_top_left
tff(fact_522_sup__top__left,axiom,
    ! [X: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,top_to1996260823553986621t_unit),X) = top_to1996260823553986621t_unit ).

% sup_top_left
tff(fact_523_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_524_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_525_inf__sup__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = X ).

% inf_sup_absorb
tff(fact_526_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = X ).

% inf_sup_absorb
tff(fact_527_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = X ).

% inf_sup_absorb
tff(fact_528_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_529_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_530_sup__inf__absorb,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = X ).

% sup_inf_absorb
tff(fact_531_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = X ).

% sup_inf_absorb
tff(fact_532_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = X ).

% sup_inf_absorb
tff(fact_533_in__range__dist__union,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,As3: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3)))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As3)) ) ) ).

% in_range_dist_union
tff(fact_534_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_left_commute
tff(fact_535_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_left_commute
tff(fact_536_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),C)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% sup.left_commute
tff(fact_537_sup_Oleft__commute,axiom,
    ! [B: assn,A: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),C)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).

% sup.left_commute
tff(fact_538_sup__commute,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% sup_commute
tff(fact_539_sup__commute,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% sup_commute
tff(fact_540_sup_Ocommute,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),A) ).

% sup.commute
tff(fact_541_sup_Ocommute,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),A) ).

% sup.commute
tff(fact_542_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% sup_assoc
tff(fact_543_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% sup_assoc
tff(fact_544_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),C) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)) ).

% sup.assoc
tff(fact_545_sup_Oassoc,axiom,
    ! [A: assn,B: assn,C: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),C) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)) ).

% sup.assoc
tff(fact_546_inf__sup__aci_I5_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X) ).

% inf_sup_aci(5)
tff(fact_547_inf__sup__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% inf_sup_aci(5)
tff(fact_548_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_549_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% inf_sup_aci(6)
tff(fact_550_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% inf_sup_aci(7)
tff(fact_551_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% inf_sup_aci(7)
tff(fact_552_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) ).

% inf_sup_aci(8)
tff(fact_553_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% inf_sup_aci(8)
tff(fact_554_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y2),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X2),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_555_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y2),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X2),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_556_distrib__imp1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y2),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),Y2)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X2),Z2))
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ) ) ).

% distrib_imp1
tff(fact_557_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X2),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ) ) ).

% distrib_imp1
tff(fact_558_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y2),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X2),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ) ) ).

% distrib_imp1
tff(fact_559_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y2),Z2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Y2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X2),Z2))
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_560_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y2),Z2)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Y2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X2),Z2))
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_561_distrib__imp2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y2),Z2)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),Y2)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X2),Z2))
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ) ) ).

% distrib_imp2
tff(fact_562_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y2),Z2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Y2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X2),Z2))
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ) ) ).

% distrib_imp2
tff(fact_563_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y2),Z2)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Y2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X2),Z2))
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ) ) ).

% distrib_imp2
tff(fact_564_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_565_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_566_inf__sup__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ).

% inf_sup_distrib1
tff(fact_567_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% inf_sup_distrib1
tff(fact_568_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% inf_sup_distrib1
tff(fact_569_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_570_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_571_inf__sup__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z),X)) ).

% inf_sup_distrib2
tff(fact_572_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).

% inf_sup_distrib2
tff(fact_573_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% inf_sup_distrib2
tff(fact_574_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_575_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_576_sup__inf__distrib1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ).

% sup_inf_distrib1
tff(fact_577_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% sup_inf_distrib1
tff(fact_578_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% sup_inf_distrib1
tff(fact_579_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_580_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)),X) = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),X)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_581_sup__inf__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z),X)) ).

% sup_inf_distrib2
tff(fact_582_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).

% sup_inf_distrib2
tff(fact_583_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% sup_inf_distrib2
tff(fact_584_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_585_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_586_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_587_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z)) ).

% boolean_algebra.conj_disj_distrib
tff(fact_588_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_589_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_590_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_591_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% boolean_algebra.disj_conj_distrib
tff(fact_592_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_593_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_594_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_595_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Z),X)) ).

% boolean_algebra.conj_disj_distrib2
tff(fact_596_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)),X) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_597_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)),X) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_598_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)),X) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_599_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)),X) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Z),X)) ).

% boolean_algebra.disj_conj_distrib2
tff(fact_600_sup_Oabel__semigroup__axioms,axiom,
    abel_s3907380067910933104et_nat(sup_sup_set_nat) ).

% sup.abel_semigroup_axioms
tff(fact_601_sup_Oabel__semigroup__axioms,axiom,
    abel_semigroup_assn(sup_sup_assn) ).

% sup.abel_semigroup_axioms
tff(fact_602_sup_Osemilattice__axioms,axiom,
    semilattice_set_nat(sup_sup_set_nat) ).

% sup.semilattice_axioms
tff(fact_603_sup_Osemilattice__axioms,axiom,
    semilattice_assn(sup_sup_assn) ).

% sup.semilattice_axioms
tff(fact_604_sup_Osemigroup__axioms,axiom,
    semigroup_set_nat(sup_sup_set_nat) ).

% sup.semigroup_axioms
tff(fact_605_sup_Osemigroup__axioms,axiom,
    semigroup_assn(sup_sup_assn) ).

% sup.semigroup_axioms
tff(fact_606_sup__bot_Ocomm__monoid__axioms,axiom,
    aa_assn_o(comm_monoid_assn(sup_sup_assn),bot_bot_assn) ).

% sup_bot.comm_monoid_axioms
tff(fact_607_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_nat(sup_sup_set_nat,bot_bot_set_nat) ).

% sup_bot.comm_monoid_axioms
tff(fact_608_sup__bot_Ocomm__monoid__axioms,axiom,
    comm_monoid_set_int(sup_sup_set_int,bot_bot_set_int) ).

% sup_bot.comm_monoid_axioms
tff(fact_609_sup__bot_Omonoid__axioms,axiom,
    aa_assn_o(monoid_assn(sup_sup_assn),bot_bot_assn) ).

% sup_bot.monoid_axioms
tff(fact_610_sup__bot_Omonoid__axioms,axiom,
    monoid_set_nat(sup_sup_set_nat,bot_bot_set_nat) ).

% sup_bot.monoid_axioms
tff(fact_611_sup__bot_Omonoid__axioms,axiom,
    monoid_set_int(sup_sup_set_int,bot_bot_set_int) ).

% sup_bot.monoid_axioms
tff(fact_612_sup__bot_Osemilattice__neutr__axioms,axiom,
    aa_assn_o(semila5388846958306562758r_assn(sup_sup_assn),bot_bot_assn) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_613_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila1241773964035338532et_nat(sup_sup_set_nat,bot_bot_set_nat) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_614_sup__bot_Osemilattice__neutr__axioms,axiom,
    semila6287294981380917632et_int(sup_sup_set_int,bot_bot_set_int) ).

% sup_bot.semilattice_neutr_axioms
tff(fact_615_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),X) = bot_bo2099793752762293965at_nat )
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),X) = top_to4669805908274784177at_nat )
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),Y) = bot_bo2099793752762293965at_nat )
         => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A),Y) = top_to4669805908274784177at_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_616_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),X) = bot_bo3516217578726220278_int_o )
     => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,A),X) = top_to8487252020945707026_int_o )
       => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),Y) = bot_bo3516217578726220278_int_o )
         => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,A),Y) = top_to8487252020945707026_int_o )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_617_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_nat,X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),X) = bot_bot_set_nat )
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),X) = top_top_set_nat )
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),Y) = bot_bot_set_nat )
         => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),Y) = top_top_set_nat )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_618_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_int,X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),X) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),X) = top_top_set_int )
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),Y) = bot_bot_set_int )
         => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,A),Y) = top_top_set_int )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_619_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: assn,X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),X) = bot_bot_assn )
     => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),X) = top_top_assn )
       => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),Y) = bot_bot_assn )
         => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),Y) = top_top_assn )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_620_boolean__algebra_Ocomplement__unique,axiom,
    ! [A: set_Product_unit,X: set_Product_unit,Y: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),X) = bot_bo3957492148770167129t_unit )
     => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),X) = top_to1996260823553986621t_unit )
       => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A),Y) = bot_bo3957492148770167129t_unit )
         => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,A),Y) = top_to1996260823553986621t_unit )
           => ( X = Y ) ) ) ) ) ).

% boolean_algebra.complement_unique
tff(fact_621_one__assn__raw_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,produc7507926704131184380et_nat(H2,As2))
    <=> ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
tff(fact_622_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ( (Y)
            <=> ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
tff(fact_623_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
tff(fact_624_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = T ).

% Int_Un_eq(4)
tff(fact_625_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = T ).

% Int_Un_eq(4)
tff(fact_626_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)) = S ).

% Int_Un_eq(3)
tff(fact_627_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)) = S ).

% Int_Un_eq(3)
tff(fact_628_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),T) = T ).

% Int_Un_eq(2)
tff(fact_629_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),T) = T ).

% Int_Un_eq(2)
tff(fact_630_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),T)),S) = S ).

% Int_Un_eq(1)
tff(fact_631_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),T)),S) = S ).

% Int_Un_eq(1)
tff(fact_632_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,T),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = T ).

% Un_Int_eq(4)
tff(fact_633_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,T),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = T ).

% Un_Int_eq(4)
tff(fact_634_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,S),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)) = S ).

% Un_Int_eq(3)
tff(fact_635_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,S),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)) = S ).

% Un_Int_eq(3)
tff(fact_636_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),T) = T ).

% Un_Int_eq(2)
tff(fact_637_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),T) = T ).

% Un_Int_eq(2)
tff(fact_638_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,S),T)),S) = S ).

% Un_Int_eq(1)
tff(fact_639_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,S),T)),S) = S ).

% Un_Int_eq(1)
tff(fact_640_wand__assnI,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat,Qa: assn,R: assn] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
     => ( ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
            ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As4) = bot_bot_set_nat )
           => ( relH(As2,H2,H4)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As2))
               => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),produc7507926704131184380et_nat(H4,As4))
                 => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,R),produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As4))) ) ) ) )
       => aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Qa),R)),produc7507926704131184380et_nat(H2,As2)) ) ) ).

% wand_assnI
tff(fact_641_IntI,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(C,A2)
     => ( member7544390268018397336_nat_o(C,B2)
       => member7544390268018397336_nat_o(C,inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% IntI
tff(fact_642_IntI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( member_int2(C,A2)
     => ( member_int2(C,B2)
       => member_int2(C,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ) ).

% IntI
tff(fact_643_IntI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,A2)
     => ( member8440522571783428010at_nat(C,B2)
       => member8440522571783428010at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% IntI
tff(fact_644_IntI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(C,A2)
     => ( member_nat2(C,B2)
       => member_nat2(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% IntI
tff(fact_645_Int__iff,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(C,inf_in3376754634429691197_nat_o(A2,B2))
    <=> ( member7544390268018397336_nat_o(C,A2)
        & member7544390268018397336_nat_o(C,B2) ) ) ).

% Int_iff
tff(fact_646_Int__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( member_int2(C,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
    <=> ( member_int2(C,A2)
        & member_int2(C,B2) ) ) ).

% Int_iff
tff(fact_647_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( member8440522571783428010at_nat(C,A2)
        & member8440522571783428010at_nat(C,B2) ) ) ).

% Int_iff
tff(fact_648_Int__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( member_nat2(C,A2)
        & member_nat2(C,B2) ) ) ).

% Int_iff
tff(fact_649_Int__UNIV,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = top_to4669805908274784177at_nat )
    <=> ( ( A2 = top_to4669805908274784177at_nat )
        & ( B2 = top_to4669805908274784177at_nat ) ) ) ).

% Int_UNIV
tff(fact_650_Int__UNIV,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = top_top_set_nat )
    <=> ( ( A2 = top_top_set_nat )
        & ( B2 = top_top_set_nat ) ) ) ).

% Int_UNIV
tff(fact_651_Int__UNIV,axiom,
    ! [A2: set_Product_unit,B2: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A2),B2) = top_to1996260823553986621t_unit )
    <=> ( ( A2 = top_to1996260823553986621t_unit )
        & ( B2 = top_to1996260823553986621t_unit ) ) ) ).

% Int_UNIV
tff(fact_652_mod__or__dist,axiom,
    ! [Pa: assn,Qa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),H2)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),H2) ) ) ).

% mod_or_dist
tff(fact_653_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,pure_assn((A))),pure_assn((B))) = pure_assn(( (A)
        | (B) )) ).

% merge_pure_or
tff(fact_654_relH__dist__union,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( relH(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As3),H2,H5)
    <=> ( relH(As2,H2,H5)
        & relH(As3,H2,H5) ) ) ).

% relH_dist_union
tff(fact_655_mod__h__bot__iff_I7_J,axiom,
    ! [Pa: assn,Qa: assn,H2: heap_e7401611519738050253t_unit] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,bot_bot_set_nat))
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Qa),produc7507926704131184380et_nat(H2,bot_bot_set_nat)) ) ) ).

% mod_h_bot_iff(7)
tff(fact_656_relH__sym,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H5)
     => relH(As2,H5,H2) ) ).

% relH_sym
tff(fact_657_relH__trans,axiom,
    ! [As2: set_nat,H12: heap_e7401611519738050253t_unit,H23: heap_e7401611519738050253t_unit,H32: heap_e7401611519738050253t_unit] :
      ( relH(As2,H12,H23)
     => ( relH(As2,H23,H32)
       => relH(As2,H12,H32) ) ) ).

% relH_trans
tff(fact_658_star__or__dist2,axiom,
    ! [C2: assn,A2: assn,B2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),A2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,C2),B2)) ).

% star_or_dist2
tff(fact_659_star__or__dist1,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C2) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A2),C2)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B2),C2)) ).

% star_or_dist1
tff(fact_660_ent__disjI2__direct,axiom,
    ! [B2: assn,A2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% ent_disjI2_direct
tff(fact_661_ent__disjI1__direct,axiom,
    ! [A2: assn,B2: assn] : aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% ent_disjI1_direct
tff(fact_662_ent__disjI2_H,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C2)) ) ).

% ent_disjI2'
tff(fact_663_ent__disjI1_H,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),B2)
     => aa_assn_o(aa_assn_fun_assn_o(entails,A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C2)) ) ).

% ent_disjI1'
tff(fact_664_ent__disjI2,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Qa),R) ) ).

% ent_disjI2
tff(fact_665_ent__disjI1,axiom,
    ! [Pa: assn,Qa: assn,R: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa)),R)
     => aa_assn_o(aa_assn_fun_assn_o(entails,Pa),R) ) ).

% ent_disjI1
tff(fact_666_ent__disjE,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(entails,A2),C2)
     => ( aa_assn_o(aa_assn_fun_assn_o(entails,B2),C2)
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C2) ) ) ).

% ent_disjE
tff(fact_667_entt__disjE,axiom,
    ! [A2: assn,M: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),M)
     => ( aa_assn_o(entailst(B2),M)
       => aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),M) ) ) ).

% entt_disjE
tff(fact_668_entt__disjD1,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C2)
     => aa_assn_o(entailst(A2),C2) ) ).

% entt_disjD1
tff(fact_669_entt__disjD2,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(entailst(aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)),C2)
     => aa_assn_o(entailst(B2),C2) ) ).

% entt_disjD2
tff(fact_670_entt__disjI1_H,axiom,
    ! [A2: assn,B2: assn,C2: assn] :
      ( aa_assn_o(entailst(A2),B2)
     => aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C2)) ) ).

% entt_disjI1'
tff(fact_671_entt__disjI2_H,axiom,
    ! [A2: assn,C2: assn,B2: assn] :
      ( aa_assn_o(entailst(A2),C2)
     => aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B2),C2)) ) ).

% entt_disjI2'
tff(fact_672_entt__disjI1__direct,axiom,
    ! [A2: assn,B2: assn] : aa_assn_o(entailst(A2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% entt_disjI1_direct
tff(fact_673_entt__disjI2__direct,axiom,
    ! [B2: assn,A2: assn] : aa_assn_o(entailst(B2),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A2),B2)) ).

% entt_disjI2_direct
tff(fact_674_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,top_top_assn),X) = top_top_assn ).

% norm_assertion_simps(11)
tff(fact_675_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),top_top_assn) = top_top_assn ).

% norm_assertion_simps(12)
tff(fact_676_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,bot_bot_assn),X) = X ).

% norm_assertion_simps(5)
tff(fact_677_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),bot_bot_assn) = X ).

% norm_assertion_simps(6)
tff(fact_678_mod__relH,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,Pa: assn] :
      ( relH(As2,H2,H5)
     => ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H2,As2))
      <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),produc7507926704131184380et_nat(H5,As2)) ) ) ).

% mod_relH
tff(fact_679_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H5)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As2)) ) ).

% relH_in_rangeI(2)
tff(fact_680_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H5)
     => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2)) ) ).

% relH_in_rangeI(1)
tff(fact_681_relH__refl,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
     => relH(As2,H2,H2) ) ).

% relH_refl
tff(fact_682_Int__UNIV__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),top_to4669805908274784177at_nat) = A2 ).

% Int_UNIV_right
tff(fact_683_Int__UNIV__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),top_top_set_nat) = A2 ).

% Int_UNIV_right
tff(fact_684_Int__UNIV__right,axiom,
    ! [A2: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,A2),top_to1996260823553986621t_unit) = A2 ).

% Int_UNIV_right
tff(fact_685_Int__UNIV__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat),B2) = B2 ).

% Int_UNIV_left
tff(fact_686_Int__UNIV__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,top_top_set_nat),B2) = B2 ).

% Int_UNIV_left
tff(fact_687_Int__UNIV__left,axiom,
    ! [B2: set_Product_unit] : aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit),B2) = B2 ).

% Int_UNIV_left
tff(fact_688_Int__left__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)) ).

% Int_left_commute
tff(fact_689_Int__left__commute,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ).

% Int_left_commute
tff(fact_690_Int__left__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% Int_left_absorb
tff(fact_691_Int__left__absorb,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Int_left_absorb
tff(fact_692_Int__commute,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) ).

% Int_commute
tff(fact_693_Int__commute,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) ).

% Int_commute
tff(fact_694_Int__absorb,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_695_Int__absorb,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),A2) = A2 ).

% Int_absorb
tff(fact_696_Int__assoc,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% Int_assoc
tff(fact_697_Int__assoc,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Int_assoc
tff(fact_698_IntD2,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(C,inf_in3376754634429691197_nat_o(A2,B2))
     => member7544390268018397336_nat_o(C,B2) ) ).

% IntD2
tff(fact_699_IntD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( member_int2(C,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => member_int2(C,B2) ) ).

% IntD2
tff(fact_700_IntD2,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => member8440522571783428010at_nat(C,B2) ) ).

% IntD2
tff(fact_701_IntD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => member_nat2(C,B2) ) ).

% IntD2
tff(fact_702_IntD1,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(C,inf_in3376754634429691197_nat_o(A2,B2))
     => member7544390268018397336_nat_o(C,A2) ) ).

% IntD1
tff(fact_703_IntD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( member_int2(C,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => member_int2(C,A2) ) ).

% IntD1
tff(fact_704_IntD1,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => member8440522571783428010at_nat(C,A2) ) ).

% IntD1
tff(fact_705_IntD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => member_nat2(C,A2) ) ).

% IntD1
tff(fact_706_IntE,axiom,
    ! [C: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(C,inf_in3376754634429691197_nat_o(A2,B2))
     => ~ ( member7544390268018397336_nat_o(C,A2)
         => ~ member7544390268018397336_nat_o(C,B2) ) ) ).

% IntE
tff(fact_707_IntE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( member_int2(C,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2))
     => ~ ( member_int2(C,A2)
         => ~ member_int2(C,B2) ) ) ).

% IntE
tff(fact_708_IntE,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(C,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
     => ~ ( member8440522571783428010at_nat(C,A2)
         => ~ member8440522571783428010at_nat(C,B2) ) ) ).

% IntE
tff(fact_709_IntE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(C,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
     => ~ ( member_nat2(C,A2)
         => ~ member_nat2(C,B2) ) ) ).

% IntE
tff(fact_710_disjoint__iff__not__equal,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,A2)
         => ! [Xa2: product_prod_nat_nat] :
              ( member8440522571783428010at_nat(Xa2,B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_711_disjoint__iff__not__equal,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( member_nat2(X4,A2)
         => ! [Xa2: nat] :
              ( member_nat2(Xa2,B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_712_disjoint__iff__not__equal,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( member_int2(X4,A2)
         => ! [Xa2: int] :
              ( member_int2(Xa2,B2)
             => ( X4 != Xa2 ) ) ) ) ).

% disjoint_iff_not_equal
tff(fact_713_Int__empty__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),bot_bo2099793752762293965at_nat) = bot_bo2099793752762293965at_nat ).

% Int_empty_right
tff(fact_714_Int__empty__right,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),bot_bot_set_nat) = bot_bot_set_nat ).

% Int_empty_right
tff(fact_715_Int__empty__right,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),bot_bot_set_int) = bot_bot_set_int ).

% Int_empty_right
tff(fact_716_Int__empty__left,axiom,
    ! [B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bot_bo2099793752762293965at_nat),B2) = bot_bo2099793752762293965at_nat ).

% Int_empty_left
tff(fact_717_Int__empty__left,axiom,
    ! [B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,bot_bot_set_nat),B2) = bot_bot_set_nat ).

% Int_empty_left
tff(fact_718_Int__empty__left,axiom,
    ! [B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,bot_bot_set_int),B2) = bot_bot_set_int ).

% Int_empty_left
tff(fact_719_disjoint__iff,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o )
    <=> ! [X4: fun_Pr6460944699146269935_nat_o] :
          ( member7544390268018397336_nat_o(X4,A2)
         => ~ member7544390268018397336_nat_o(X4,B2) ) ) ).

% disjoint_iff
tff(fact_720_disjoint__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> ! [X4: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X4,A2)
         => ~ member8440522571783428010at_nat(X4,B2) ) ) ).

% disjoint_iff
tff(fact_721_disjoint__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> ! [X4: nat] :
          ( member_nat2(X4,A2)
         => ~ member_nat2(X4,B2) ) ) ).

% disjoint_iff
tff(fact_722_disjoint__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> ! [X4: int] :
          ( member_int2(X4,A2)
         => ~ member_int2(X4,B2) ) ) ).

% disjoint_iff
tff(fact_723_Int__emptyI,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( member7544390268018397336_nat_o(X2,A2)
         => ~ member7544390268018397336_nat_o(X2,B2) )
     => ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ).

% Int_emptyI
tff(fact_724_Int__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X2,A2)
         => ~ member8440522571783428010at_nat(X2,B2) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
tff(fact_725_Int__emptyI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X2: nat] :
          ( member_nat2(X2,A2)
         => ~ member_nat2(X2,B2) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% Int_emptyI
tff(fact_726_Int__emptyI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X2: int] :
          ( member_int2(X2,A2)
         => ~ member_int2(X2,B2) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ).

% Int_emptyI
tff(fact_727_Un__Int__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A2)) ).

% Un_Int_distrib2
tff(fact_728_Un__Int__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ).

% Un_Int_distrib2
tff(fact_729_Int__Un__distrib2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)),A2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)) ).

% Int_Un_distrib2
tff(fact_730_Int__Un__distrib2,axiom,
    ! [B2: set_nat,C2: set_nat,A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)),A2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) ).

% Int_Un_distrib2
tff(fact_731_Un__Int__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),C2)) ).

% Un_Int_distrib
tff(fact_732_Un__Int__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),C2)) ).

% Un_Int_distrib
tff(fact_733_Int__Un__distrib,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)) ).

% Int_Un_distrib
tff(fact_734_Int__Un__distrib,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)) ).

% Int_Un_distrib
tff(fact_735_Un__Int__crazy,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,C2),A2)) ).

% Un_Int_crazy
tff(fact_736_Un__Int__crazy,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C2),A2)) ).

% Un_Int_crazy
tff(fact_737_wand__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
              & ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
                  ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As4) = bot_bot_set_nat )
                    & relH(As,H3,H4)
                    & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As4)) )
                 => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4))) ) ) ) ) ).

% wand_raw.elims(3)
tff(fact_738_wand__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                    ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                      & relH(As,H3,H6)
                      & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
                   => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ).

% wand_raw.elims(2)
tff(fact_739_wand__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
      <=> (Y) )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ( (Y)
            <=> ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                  & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                        & relH(As,H3,H7)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) ) ) ) ).

% wand_raw.elims(1)
tff(fact_740_wand__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(wand_raw(Pa,Qa),produc7507926704131184380et_nat(H2,As2))
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
        & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
            ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As2),As6) = bot_bot_set_nat )
              & relH(As2,H2,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As2))
              & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H7,As6)) )
           => aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As2),As6))) ) ) ) ).

% wand_raw.simps
tff(fact_741_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As12))
                & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As23)) ) ) ) ).

% times_assn_raw.elims(3)
tff(fact_742_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                  & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                  & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As13))
                  & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As24)) ) ) ) ).

% times_assn_raw.elims(2)
tff(fact_743_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
      <=> (Y) )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb = produc7507926704131184380et_nat(H3,As) )
           => ( (Y)
            <=> ~ ? [As1: set_nat,As22: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As1))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As22)) ) ) ) ) ).

% times_assn_raw.elims(1)
tff(fact_744_times__assn__raw_Osimps,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(times_assn_raw(Pa,Qa),produc7507926704131184380et_nat(H2,As2))
    <=> ? [As1: set_nat,As22: set_nat] :
          ( ( As2 = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
          & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
          & aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As1))
          & aa_Pro653082453696054616_nat_o(Qa,produc7507926704131184380et_nat(H2,As22)) ) ) ).

% times_assn_raw.simps
tff(fact_745_disjointI,axiom,
    ! [A: set_fu3555266933746557007_nat_o,B: set_fu3555266933746557007_nat_o] :
      ( ! [X2: fun_Pr6460944699146269935_nat_o] :
          ( member7544390268018397336_nat_o(X2,A)
         => ~ member7544390268018397336_nat_o(X2,B) )
     => ( inf_in3376754634429691197_nat_o(A,B) = bot_bo8632255151191677115_nat_o ) ) ).

% disjointI
tff(fact_746_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( member8440522571783428010at_nat(X2,A)
         => ~ member8440522571783428010at_nat(X2,B) )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
tff(fact_747_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( member_nat2(X2,A)
         => ~ member_nat2(X2,B) )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ).

% disjointI
tff(fact_748_disjointI,axiom,
    ! [A: set_int,B: set_int] :
      ( ! [X2: int] :
          ( member_int2(X2,A)
         => ~ member_int2(X2,B) )
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ).

% disjointI
tff(fact_749_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y) = top_to4669805908274784177at_nat )
       => ( aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_750_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = bot_bo3516217578726220278_int_o )
     => ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y) = top_to8487252020945707026_int_o )
       => ( aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_751_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = top_top_set_nat )
       => ( aa_set_nat_set_nat(uminus5710092332889474511et_nat,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_752_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
     => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(sup_sup_set_int,X),Y) = top_top_set_int )
       => ( aa_set_int_set_int(uminus1532241313380277803et_int,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_753_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: set_Product_unit,Y: set_Product_unit] :
      ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(inf_in4660618365625256667t_unit,X),Y) = bot_bo3957492148770167129t_unit )
     => ( ( aa_set8324764978855220220t_unit(aa_set170811390870066356t_unit(sup_su793286257634532545t_unit,X),Y) = top_to1996260823553986621t_unit )
       => ( aa_set8324764978855220220t_unit(uminus5944136376168626660t_unit,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_754_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
     => ( ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = top_top_assn )
       => ( aa_assn_assn(uminus_uminus_assn,X) = Y ) ) ) ).

% boolean_algebra_class.boolean_algebra.compl_unique
tff(fact_755_properI,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( ! [As: set_nat,H3: heap_e7401611519738050253t_unit] :
          ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H3,As))
         => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As)) )
     => ( ! [As: set_nat,H3: heap_e7401611519738050253t_unit,H4: heap_e7401611519738050253t_unit] :
            ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H3,As))
           => ( relH(As,H3,H4)
             => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
               => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H4,As)) ) ) )
       => aa_fun1675921005893230629at_o_o(proper,Pa) ) ) ).

% properI
tff(fact_756_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = aa_int_int(uminus_uminus_int,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_757_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = uminus_uminus_rat(B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_758_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( A = B ) ) ).

% neg_equal_iff_equal
tff(fact_759_add_Oinverse__inverse,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,aa_int_int(uminus_uminus_int,A)) = A ).

% add.inverse_inverse
tff(fact_760_add_Oinverse__inverse,axiom,
    ! [A: rat] : uminus_uminus_rat(uminus_uminus_rat(A)) = A ).

% add.inverse_inverse
tff(fact_761_add_Oinverse__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = A ).

% add.inverse_inverse
tff(fact_762_pure__assn__proper,axiom,
    ! [B: $o] : aa_fun1675921005893230629at_o_o(proper,pure_a825153325127701367it_nat((B))) ).

% pure_assn_proper
tff(fact_763_Compl__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint
tff(fact_764_Compl__disjoint,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)) = bot_bot_set_nat ).

% Compl_disjoint
tff(fact_765_Compl__disjoint,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),aa_set_int_set_int(uminus1532241313380277803et_int,A2)) = bot_bot_set_int ).

% Compl_disjoint
tff(fact_766_Compl__disjoint2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),A2) = bot_bo2099793752762293965at_nat ).

% Compl_disjoint2
tff(fact_767_Compl__disjoint2,axiom,
    ! [A2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),A2) = bot_bot_set_nat ).

% Compl_disjoint2
tff(fact_768_Compl__disjoint2,axiom,
    ! [A2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,A2)),A2) = bot_bot_set_int ).

% Compl_disjoint2
tff(fact_769_mod__not__dist,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(uminus_uminus_assn,Pa)),H2)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,H2)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H2) ) ) ).

% mod_not_dist
tff(fact_770_bool__assn__proper_I1_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,in_range) ).

% bool_assn_proper(1)
tff(fact_771_times__assn__proper,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Qa)
       => aa_fun1675921005893230629at_o_o(proper,times_assn_raw(Pa,Qa)) ) ) ).

% times_assn_proper
tff(fact_772_wand__proper,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o] : aa_fun1675921005893230629at_o_o(proper,wand_raw(Pa,Qa)) ).

% wand_proper
tff(fact_773_one__assn__proper,axiom,
    aa_fun1675921005893230629at_o_o(proper,one_assn_raw) ).

% one_assn_proper
tff(fact_774_inf__compl__bot__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left1
tff(fact_775_inf__compl__bot__left1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = bot_bo3516217578726220278_int_o ).

% inf_compl_bot_left1
tff(fact_776_inf__compl__bot__left1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left1
tff(fact_777_inf__compl__bot__left1,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y)) = bot_bot_set_int ).

% inf_compl_bot_left1
tff(fact_778_inf__compl__bot__left1,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = bot_bot_assn ).

% inf_compl_bot_left1
tff(fact_779_inf__compl__bot__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),Y)) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_left2
tff(fact_780_inf__compl__bot__left2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),Y)) = bot_bo3516217578726220278_int_o ).

% inf_compl_bot_left2
tff(fact_781_inf__compl__bot__left2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),Y)) = bot_bot_set_nat ).

% inf_compl_bot_left2
tff(fact_782_inf__compl__bot__left2,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),Y)) = bot_bot_set_int ).

% inf_compl_bot_left2
tff(fact_783_inf__compl__bot__left2,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),Y)) = bot_bot_assn ).

% inf_compl_bot_left2
tff(fact_784_inf__compl__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X))) = bot_bo2099793752762293965at_nat ).

% inf_compl_bot_right
tff(fact_785_inf__compl__bot__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X))) = bot_bo3516217578726220278_int_o ).

% inf_compl_bot_right
tff(fact_786_inf__compl__bot__right,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X))) = bot_bot_set_nat ).

% inf_compl_bot_right
tff(fact_787_inf__compl__bot__right,axiom,
    ! [X: set_int,Y: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,Y),aa_set_int_set_int(uminus1532241313380277803et_int,X))) = bot_bot_set_int ).

% inf_compl_bot_right
tff(fact_788_inf__compl__bot__right,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),aa_assn_assn(uminus_uminus_assn,X))) = bot_bot_assn ).

% inf_compl_bot_right
tff(fact_789_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),X) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_790_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),X) = bot_bo3516217578726220278_int_o ).

% boolean_algebra.conj_cancel_left
tff(fact_791_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),X) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_left
tff(fact_792_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),X) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_left
tff(fact_793_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),X) = bot_bot_assn ).

% boolean_algebra.conj_cancel_left
tff(fact_794_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)) = bot_bo2099793752762293965at_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_795_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)) = bot_bo3516217578726220278_int_o ).

% boolean_algebra.conj_cancel_right
tff(fact_796_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)) = bot_bot_set_nat ).

% boolean_algebra.conj_cancel_right
tff(fact_797_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),aa_set_int_set_int(uminus1532241313380277803et_int,X)) = bot_bot_set_int ).

% boolean_algebra.conj_cancel_right
tff(fact_798_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,X)) = bot_bot_assn ).

% boolean_algebra.conj_cancel_right
tff(fact_799_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_800_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_801_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_802_boolean__algebra_Ode__Morgan__conj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_conj
tff(fact_803_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_804_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_805_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_806_boolean__algebra_Ode__Morgan__disj,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(uminus_uminus_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),aa_assn_assn(uminus_uminus_assn,Y)) ).

% boolean_algebra.de_Morgan_disj
tff(fact_807_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) ).

% assn_aci(8)
tff(fact_808_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z)) ).

% assn_aci(7)
tff(fact_809_assn__aci_I5_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),X) ).

% assn_aci(5)
tff(fact_810_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),X) = X ).

% norm_assertion_simps(32)
tff(fact_811_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z) = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ).

% norm_assertion_simps(15)
tff(fact_812_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(uminus_uminus_int,B) = A ) ) ).

% minus_equation_iff
tff(fact_813_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( uminus_uminus_rat(B) = A ) ) ).

% minus_equation_iff
tff(fact_814_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) = A ) ) ).

% minus_equation_iff
tff(fact_815_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% equation_minus_iff
tff(fact_816_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% equation_minus_iff
tff(fact_817_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% equation_minus_iff
tff(fact_818_Rep__assn__induct,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o,Pa: fun_fu5135969500543783852at_o_o] :
      ( member7544390268018397336_nat_o(Y,collec6407654713642994138_nat_o(proper))
     => ( ! [X2: assn] : aa_fun1675921005893230629at_o_o(Pa,aa_ass4243947775541419017_nat_o(rep_assn,X2))
       => aa_fun1675921005893230629at_o_o(Pa,Y) ) ) ).

% Rep_assn_induct
tff(fact_819_Rep__assn__cases,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( member7544390268018397336_nat_o(Y,collec6407654713642994138_nat_o(proper))
     => ~ ! [X2: assn] : Y != aa_ass4243947775541419017_nat_o(rep_assn,X2) ) ).

% Rep_assn_cases
tff(fact_820_Rep__assn,axiom,
    ! [X: assn] : member7544390268018397336_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,X),collec6407654713642994138_nat_o(proper)) ).

% Rep_assn
tff(fact_821_inf__cancel__left1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left1
tff(fact_822_inf__cancel__left1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),A)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),B)) = bot_bo3516217578726220278_int_o ).

% inf_cancel_left1
tff(fact_823_inf__cancel__left1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),B)) = bot_bot_set_nat ).

% inf_cancel_left1
tff(fact_824_inf__cancel__left1,axiom,
    ! [X: set_int,A: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),B)) = bot_bot_set_int ).

% inf_cancel_left1
tff(fact_825_inf__cancel__left1,axiom,
    ! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),B)) = bot_bot_assn ).

% inf_cancel_left1
tff(fact_826_inf__cancel__left2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,X)),A)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),B)) = bot_bo2099793752762293965at_nat ).

% inf_cancel_left2
tff(fact_827_inf__cancel__left2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,X)),A)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),B)) = bot_bo3516217578726220278_int_o ).

% inf_cancel_left2
tff(fact_828_inf__cancel__left2,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,X)),A)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),B)) = bot_bot_set_nat ).

% inf_cancel_left2
tff(fact_829_inf__cancel__left2,axiom,
    ! [X: set_int,A: set_int,B: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(uminus1532241313380277803et_int,X)),A)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),B)) = bot_bot_set_int ).

% inf_cancel_left2
tff(fact_830_inf__cancel__left2,axiom,
    ! [X: assn,A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(uminus_uminus_assn,X)),A)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),B)) = bot_bot_assn ).

% inf_cancel_left2
tff(fact_831_Compl__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ).

% Compl_Un
tff(fact_832_Compl__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ).

% Compl_Un
tff(fact_833_Compl__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,A2)),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ).

% Compl_Int
tff(fact_834_Compl__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(uminus5710092332889474511et_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,A2)),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ).

% Compl_Int
tff(fact_835_properD1,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As2))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2)) ) ) ).

% properD1
tff(fact_836_proper__iff,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( relH(As2,H2,H5)
       => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As2))
         => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As2))
          <=> aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H5,As2)) ) ) ) ) ).

% proper_iff
tff(fact_837_proper__def,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
    <=> ! [H: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As7: set_nat] :
          ( ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As7))
           => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H,As7)) )
          & ( ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H,As7))
              & relH(As7,H,H7)
              & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As7)) )
           => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H7,As7)) ) ) ) ).

% proper_def
tff(fact_838_properD2,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,H2: heap_e7401611519738050253t_unit,As2: set_nat,H5: heap_e7401611519738050253t_unit] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H2,As2))
       => ( relH(As2,H2,H5)
         => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As2))
           => aa_Pro653082453696054616_nat_o(Pa,produc7507926704131184380et_nat(H5,As2)) ) ) ) ) ).

% properD2
tff(fact_839_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_840_mult__minus1,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(one_one_rat)),Z) = uminus_uminus_rat(Z) ).

% mult_minus1
tff(fact_841_mult__minus1,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),Z) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1
tff(fact_842_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_843_mult__minus1__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(Z) ).

% mult_minus1_right
tff(fact_844_mult__minus1__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Z) ).

% mult_minus1_right
tff(fact_845_mult__minus__left,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_left
tff(fact_846_mult__minus__left,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),B) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_left
tff(fact_847_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_left
tff(fact_848_minus__mult__minus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),B) ).

% minus_mult_minus
tff(fact_849_minus__mult__minus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),uminus_uminus_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) ).

% minus_mult_minus
tff(fact_850_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) ).

% minus_mult_minus
tff(fact_851_mult__minus__right,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% mult_minus_right
tff(fact_852_mult__minus__right,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(B)) = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ).

% mult_minus_right
tff(fact_853_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% mult_minus_right
tff(fact_854_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),X) = one_one_int )
    <=> ( ( X = one_one_int )
        | ( X = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% square_eq_1_iff
tff(fact_855_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X) = one_one_rat )
    <=> ( ( X = one_one_rat )
        | ( X = uminus_uminus_rat(one_one_rat) ) ) ) ).

% square_eq_1_iff
tff(fact_856_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X) = one_one_Code_integer )
    <=> ( ( X = one_one_Code_integer )
        | ( X = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% square_eq_1_iff
tff(fact_857_one__neq__neg__one,axiom,
    one_one_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% one_neq_neg_one
tff(fact_858_one__neq__neg__one,axiom,
    one_one_rat != uminus_uminus_rat(one_one_rat) ).

% one_neq_neg_one
tff(fact_859_one__neq__neg__one,axiom,
    one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% one_neq_neg_one
tff(fact_860_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),A) = aa_int_int(aa_int_fun_int_int(times_times_int,B),B) )
    <=> ( ( A = B )
        | ( A = aa_int_int(uminus_uminus_int,B) ) ) ) ).

% square_eq_iff
tff(fact_861_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),B) )
    <=> ( ( A = B )
        | ( A = uminus_uminus_rat(B) ) ) ) ).

% square_eq_iff
tff(fact_862_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),B) )
    <=> ( ( A = B )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ) ) ) ).

% square_eq_iff
tff(fact_863_minus__mult__commute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(uminus_uminus_int,B)) ).

% minus_mult_commute
tff(fact_864_minus__mult__commute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(B)) ).

% minus_mult_commute
tff(fact_865_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_mult_commute
tff(fact_866_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea949959532490082519at_nat(inf_in2572325071724192079at_nat,sup_su6327502436637775413at_nat,uminus6524753893492686040at_nat,bot_bo2099793752762293965at_nat,top_to4669805908274784177at_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_867_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea44587999064921452_int_o(inf_in1782598092406300404_int_o,sup_su4047618219528032398_int_o,uminus103585356917985579_int_o,bot_bo3516217578726220278_int_o,top_to8487252020945707026_int_o) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_868_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea778851993438741648et_nat(inf_inf_set_nat,sup_sup_set_nat,uminus5710092332889474511et_nat,bot_bot_set_nat,top_top_set_nat) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_869_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea5824373010784320748et_int(inf_inf_set_int,sup_sup_set_int,uminus1532241313380277803et_int,bot_bot_set_int,top_top_set_int) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_870_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    boolea3191171876098928227t_unit(inf_in4660618365625256667t_unit,sup_su793286257634532545t_unit,uminus5944136376168626660t_unit,bot_bo3957492148770167129t_unit,top_to1996260823553986621t_unit) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_871_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
    aa_assn_o(boolea5875918072483028530a_assn(inf_inf_assn,sup_sup_assn,uminus_uminus_assn,bot_bot_assn),top_top_assn) ).

% boolean_algebra.abstract_boolean_algebra_axioms
tff(fact_872_wand__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(wand_assn,Pa),Qa) = aa_fun5526454687433275355o_assn(abs_assn,wand_raw(aa_ass4243947775541419017_nat_o(rep_assn,Pa),aa_ass4243947775541419017_nat_o(rep_assn,Qa))) ).

% wand_assn_def
tff(fact_873_Rep__assn__inverse,axiom,
    ! [X: assn] : aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(rep_assn,X)) = X ).

% Rep_assn_inverse
tff(fact_874_Abs__assn__eqI_I2_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H3)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H3) )
     => ( Pr = aa_fun5526454687433275355o_assn(abs_assn,Pa) ) ) ).

% Abs_assn_eqI(2)
tff(fact_875_Abs__assn__eqI_I1_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Pr: assn] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(Pa,H3)
        <=> aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pr),H3) )
     => ( aa_fun5526454687433275355o_assn(abs_assn,Pa) = Pr ) ) ).

% Abs_assn_eqI(1)
tff(fact_876_pure__assn__def,axiom,
    ! [B: $o] : pure_assn((B)) = aa_fun5526454687433275355o_assn(abs_assn,pure_a825153325127701367it_nat((B))) ).

% pure_assn_def
tff(fact_877_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( ( X = aa_fun5526454687433275355o_assn(abs_assn,Y2) )
         => ~ member7544390268018397336_nat_o(Y2,collec6407654713642994138_nat_o(proper)) ) ).

% Abs_assn_cases
tff(fact_878_Abs__assn__induct,axiom,
    ! [Pa: fun_assn_o,X: assn] :
      ( ! [Y2: fun_Pr6460944699146269935_nat_o] :
          ( member7544390268018397336_nat_o(Y2,collec6407654713642994138_nat_o(proper))
         => aa_assn_o(Pa,aa_fun5526454687433275355o_assn(abs_assn,Y2)) )
     => aa_assn_o(Pa,X) ) ).

% Abs_assn_induct
tff(fact_879_Abs__assn__inject,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Y: fun_Pr6460944699146269935_nat_o] :
      ( member7544390268018397336_nat_o(X,collec6407654713642994138_nat_o(proper))
     => ( member7544390268018397336_nat_o(Y,collec6407654713642994138_nat_o(proper))
       => ( ( aa_fun5526454687433275355o_assn(abs_assn,X) = aa_fun5526454687433275355o_assn(abs_assn,Y) )
        <=> ( X = Y ) ) ) ) ).

% Abs_assn_inject
tff(fact_880_Abs__assn__inverse,axiom,
    ! [Y: fun_Pr6460944699146269935_nat_o] :
      ( member7544390268018397336_nat_o(Y,collec6407654713642994138_nat_o(proper))
     => ( aa_ass4243947775541419017_nat_o(rep_assn,aa_fun5526454687433275355o_assn(abs_assn,Y)) = Y ) ) ).

% Abs_assn_inverse
tff(fact_881_top__assn__def,axiom,
    top_top_assn = aa_fun5526454687433275355o_assn(abs_assn,in_range) ).

% top_assn_def
tff(fact_882_one__assn__def,axiom,
    one_one_assn = aa_fun5526454687433275355o_assn(abs_assn,one_assn_raw) ).

% one_assn_def
tff(fact_883_times__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),Qa) = aa_fun5526454687433275355o_assn(abs_assn,times_assn_raw(aa_ass4243947775541419017_nat_o(rep_assn,Pa),aa_ass4243947775541419017_nat_o(rep_assn,Qa))) ).

% times_assn_def
tff(fact_884_dbl__inc__simps_I4_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% dbl_inc_simps(4)
tff(fact_885_dbl__inc__simps_I4_J,axiom,
    neg_nu5219082963157363817nc_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% dbl_inc_simps(4)
tff(fact_886_dbl__inc__simps_I4_J,axiom,
    neg_nu5831290666863070958nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_inc_simps(4)
tff(fact_887_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_Pr5680576844133773770et_nat] : list_c6061723043370948915sn_nat(F,nil_Pr5671120429643327159n_assn) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_888_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_int_set_nat] : list_c3451693981498911948nt_nat(F,nil_int) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_889_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_nat_set_nat] : list_c2452340269597857392at_nat(F,nil_nat) = bot_bot_set_nat ).

% list_collect_set_simps(1)
tff(fact_890_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_Pr3190493699202047014et_int] : list_c6059232572861898639sn_int(F,nil_Pr5671120429643327159n_assn) = bot_bot_set_int ).

% list_collect_set_simps(1)
tff(fact_891_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_int_set_int] : list_c3449203510989861672nt_int(F,nil_int) = bot_bot_set_int ).

% list_collect_set_simps(1)
tff(fact_892_list__collect__set__simps_I1_J,axiom,
    ! [F: fun_nat_set_int] : list_c2449849799088807116at_int(F,nil_nat) = bot_bot_set_int ).

% list_collect_set_simps(1)
tff(fact_893_mod__pure,axiom,
    ! [B: $o,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,pure_assn((B))),H2)
    <=> ( ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2) = bot_bot_set_nat )
        & (B) ) ) ).

% mod_pure
tff(fact_894_type__definition__assn,axiom,
    type_d5492019815722432281_nat_o(rep_assn,abs_assn,collec6407654713642994138_nat_o(proper)) ).

% type_definition_assn
tff(fact_895_dbl__dec__simps_I3_J,axiom,
    neg_nu7757733837767384882nteger(one_one_Code_integer) = one_one_Code_integer ).

% dbl_dec_simps(3)
tff(fact_896_dbl__dec__simps_I3_J,axiom,
    neg_nu3179335615603231917ec_rat(one_one_rat) = one_one_rat ).

% dbl_dec_simps(3)
tff(fact_897_dbl__dec__simps_I3_J,axiom,
    aa_int_int(neg_nu3811975205180677377ec_int,one_one_int) = one_one_int ).

% dbl_dec_simps(3)
tff(fact_898_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord2612477271533052124et_int(bot_bot_set_int) = nil_int ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_899_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    linord2614967742042102400et_nat(bot_bot_set_nat) = nil_nat ).

% sorted_list_of_set.sorted_key_list_of_set_empty
tff(fact_900_UNIV__coset,axiom,
    top_to5722533637347664993n_assn = coset_703488708621576356n_assn(nil_Pr5671120429643327159n_assn) ).

% UNIV_coset
tff(fact_901_UNIV__coset,axiom,
    top_top_set_int = coset_int(nil_int) ).

% UNIV_coset
tff(fact_902_UNIV__coset,axiom,
    top_top_set_nat = coset_nat(nil_nat) ).

% UNIV_coset
tff(fact_903_UNIV__coset,axiom,
    top_to1996260823553986621t_unit = coset_Product_unit(nil_Product_unit) ).

% UNIV_coset
tff(fact_904_mod__emp,axiom,
    ! [H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,one_one_assn),H2)
    <=> ( aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2) = bot_bot_set_nat ) ) ).

% mod_emp
tff(fact_905_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_906_dbl__dec__simps_I2_J,axiom,
    neg_nu3179335615603231917ec_rat(zero_zero_rat) = uminus_uminus_rat(one_one_rat) ).

% dbl_dec_simps(2)
tff(fact_907_dbl__dec__simps_I2_J,axiom,
    neg_nu7757733837767384882nteger(zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% dbl_dec_simps(2)
tff(fact_908_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_int] :
      ( finite_finite_int(A2)
     => ( ( linord2612477271533052124et_int(A2) = nil_int )
      <=> ( A2 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_909_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A2: set_nat] :
      ( finite_finite_nat(A2)
     => ( ( linord2614967742042102400et_nat(A2) = nil_nat )
      <=> ( A2 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
tff(fact_910_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_nat_set_nat,X: fun_nat_nat] : list_c2452340269597857392at_nat(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),nil_nat)) = bot_bot_set_nat ).

% list_collect_set_map_simps(1)
tff(fact_911_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_assn_set_nat,X: fun_Pr3816910165196019820n_assn] : list_c1844713377658005960sn_nat(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),nil_Pr5671120429643327159n_assn)) = bot_bot_set_nat ).

% list_collect_set_map_simps(1)
tff(fact_912_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_nat_set_int,X: fun_nat_nat] : list_c2449849799088807116at_int(F,aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,X),nil_nat)) = bot_bot_set_int ).

% list_collect_set_map_simps(1)
tff(fact_913_list__collect__set__map__simps_I1_J,axiom,
    ! [F: fun_assn_set_int,X: fun_Pr3816910165196019820n_assn] : list_c1842222907148955684sn_int(F,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,X),nil_Pr5671120429643327159n_assn)) = bot_bot_set_int ).

% list_collect_set_map_simps(1)
tff(fact_914_nths__empty,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : nths_P5537322223065504949n_assn(Xs,bot_bot_set_nat) = nil_Pr5671120429643327159n_assn ).

% nths_empty
tff(fact_915_nths__empty,axiom,
    ! [Xs: list_int] : nths_int(Xs,bot_bot_set_nat) = nil_int ).

% nths_empty
tff(fact_916_nths__empty,axiom,
    ! [Xs: list_nat] : nths_nat(Xs,bot_bot_set_nat) = nil_nat ).

% nths_empty
tff(fact_917_dbl__inc__simps_I2_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,zero_zero_int) = one_one_int ).

% dbl_inc_simps(2)
tff(fact_918_dbl__inc__simps_I2_J,axiom,
    neg_nu5219082963157363817nc_rat(zero_zero_rat) = one_one_rat ).

% dbl_inc_simps(2)
tff(fact_919_dbl__inc__simps_I2_J,axiom,
    neg_nu5831290666863070958nteger(zero_z3403309356797280102nteger) = one_one_Code_integer ).

% dbl_inc_simps(2)
tff(fact_920_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_921_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
    <=> ord_less_eq_set_int(A2,aa_set_int_set_int(uminus1532241313380277803et_int,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_922_disjoint__eq__subset__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) ) ).

% disjoint_eq_subset_Compl
tff(fact_923_in__measures_I1_J,axiom,
    ! [X: nat,Y: nat] : ~ member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(nil_fun_nat_nat)) ).

% in_measures(1)
tff(fact_924_in__measures_I1_J,axiom,
    ! [X: int,Y: int] : ~ member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(nil_fun_int_nat)) ).

% in_measures(1)
tff(fact_925_sup__neg__inf,axiom,
    ! [P4: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,P4),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Q2),R2))
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,P4),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Q2))),R2) ) ).

% sup_neg_inf
tff(fact_926_sup__neg__inf,axiom,
    ! [P4: fun_Pr3238374608701176162_int_o,Q2: fun_Pr3238374608701176162_int_o,R2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,P4),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Q2),R2))
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,P4),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Q2))),R2) ) ).

% sup_neg_inf
tff(fact_927_sup__neg__inf,axiom,
    ! [P4: assn,Q2: assn,R2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,P4),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Q2),R2))
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,P4),aa_assn_assn(uminus_uminus_assn,Q2))),R2) ) ).

% sup_neg_inf
tff(fact_928_sup__neg__inf,axiom,
    ! [P4: set_nat,Q2: set_nat,R2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,P4),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Q2),R2))
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,P4),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Q2))),R2) ) ).

% sup_neg_inf
tff(fact_929_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y))),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z)) ) ).

% shunt2
tff(fact_930_shunt2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y))),Z)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z)) ) ).

% shunt2
tff(fact_931_shunt2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y))),Z)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)) ) ).

% shunt2
tff(fact_932_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y))),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)) ) ).

% shunt2
tff(fact_933_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Z)
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)),Z)) ) ).

% shunt1
tff(fact_934_shunt1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Z)
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)),Z)) ) ).

% shunt1
tff(fact_935_shunt1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Z)
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(uminus_uminus_assn,Y)),Z)) ) ).

% shunt1
tff(fact_936_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Z)
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)),Z)) ) ).

% shunt1
tff(fact_937_relH__def,axiom,
    ! [As2: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
      ( relH(As2,H2,H5)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H5,As2))
        & ! [T2: typerep,X4: nat] :
            ( member_nat2(X4,As2)
           => ( ( aa_nat_nat(refs_Product_unit(H2,T2),X4) = aa_nat_nat(refs_Product_unit(H5,T2),X4) )
              & ( aa_nat_list_nat(arrays_Product_unit(H2,T2),X4) = aa_nat_list_nat(arrays_Product_unit(H5,T2),X4) ) ) ) ) ) ).

% relH_def
tff(fact_938_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_939_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_940_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_941_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_942_mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_943_mult__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C) )
    <=> ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_right
tff(fact_944_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_945_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
    <=> ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_946_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_947_mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_948_mult__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B) )
    <=> ( ( C = zero_z2226904508553997617atural )
        | ( A = B ) ) ) ).

% mult_cancel_left
tff(fact_949_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
    <=> ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
tff(fact_950_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
    <=> ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
tff(fact_951_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
    <=> ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
tff(fact_952_mult__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% mult_eq_0_iff
tff(fact_953_mult__eq__0__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = zero_z2226904508553997617atural )
    <=> ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% mult_eq_0_iff
tff(fact_954_mult__zero__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),zero_zero_rat) = zero_zero_rat ).

% mult_zero_right
tff(fact_955_mult__zero__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),zero_zero_nat) = zero_zero_nat ).

% mult_zero_right
tff(fact_956_mult__zero__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),zero_zero_int) = zero_zero_int ).

% mult_zero_right
tff(fact_957_mult__zero__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% mult_zero_right
tff(fact_958_mult__zero__right,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),zero_z2226904508553997617atural) = zero_z2226904508553997617atural ).

% mult_zero_right
tff(fact_959_mult__zero__left,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,zero_zero_rat),A) = zero_zero_rat ).

% mult_zero_left
tff(fact_960_mult__zero__left,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_zero_nat),A) = zero_zero_nat ).

% mult_zero_left
tff(fact_961_mult__zero__left,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,zero_zero_int),A) = zero_zero_int ).

% mult_zero_left
tff(fact_962_mult__zero__left,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_z3403309356797280102nteger),A) = zero_z3403309356797280102nteger ).

% mult_zero_left
tff(fact_963_mult__zero__left,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_z2226904508553997617atural),A) = zero_z2226904508553997617atural ).

% mult_zero_left
tff(fact_964_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% neg_le_iff_le
tff(fact_965_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_eq_rat(A),B) ) ).

% neg_le_iff_le
tff(fact_966_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% neg_le_iff_le
tff(fact_967_add_Oinverse__neutral,axiom,
    aa_int_int(uminus_uminus_int,zero_zero_int) = zero_zero_int ).

% add.inverse_neutral
tff(fact_968_add_Oinverse__neutral,axiom,
    uminus_uminus_rat(zero_zero_rat) = zero_zero_rat ).

% add.inverse_neutral
tff(fact_969_add_Oinverse__neutral,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% add.inverse_neutral
tff(fact_970_neg__0__equal__iff__equal,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(uminus_uminus_int,A) )
    <=> ( zero_zero_int = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_971_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = uminus_uminus_rat(A) )
    <=> ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_972_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
tff(fact_973_neg__equal__0__iff__equal,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_0_iff_equal
tff(fact_974_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
tff(fact_975_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
tff(fact_976_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A = aa_int_int(uminus_uminus_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% equal_neg_zero
tff(fact_977_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A = uminus_uminus_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% equal_neg_zero
tff(fact_978_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
tff(fact_979_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = A )
    <=> ( A = zero_zero_int ) ) ).

% neg_equal_zero
tff(fact_980_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat(A) = A )
    <=> ( A = zero_zero_rat ) ) ).

% neg_equal_zero
tff(fact_981_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = A )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
tff(fact_982_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_983_inf_Obounded__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C) ) ) ).

% inf.bounded_iff
tff(fact_984_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).

% inf.bounded_iff
tff(fact_985_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_986_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
    <=> ( aa_rat_o(ord_less_eq_rat(A),B)
        & aa_rat_o(ord_less_eq_rat(A),C) ) ) ).

% inf.bounded_iff
tff(fact_987_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.bounded_iff
tff(fact_988_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.bounded_iff
tff(fact_989_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_990_le__inf__iff,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z))
    <=> ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
        & aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z) ) ) ).

% le_inf_iff
tff(fact_991_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z) ) ) ).

% le_inf_iff
tff(fact_992_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_993_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))
    <=> ( aa_rat_o(ord_less_eq_rat(X),Y)
        & aa_rat_o(ord_less_eq_rat(X),Z) ) ) ).

% le_inf_iff
tff(fact_994_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z) ) ) ).

% le_inf_iff
tff(fact_995_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z) ) ) ).

% le_inf_iff
tff(fact_996_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),Z)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),Z) ) ) ).

% le_sup_iff
tff(fact_997_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),Z)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_998_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),Z)
    <=> ( aa_rat_o(ord_less_eq_rat(X),Z)
        & aa_rat_o(ord_less_eq_rat(Y),Z) ) ) ).

% le_sup_iff
tff(fact_999_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1000_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),Z) ) ) ).

% le_sup_iff
tff(fact_1001_sup_Obounded__iff,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
        & aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1002_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1003_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
    <=> ( aa_rat_o(ord_less_eq_rat(B),A)
        & aa_rat_o(ord_less_eq_rat(C),A) ) ) ).

% sup.bounded_iff
tff(fact_1004_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1005_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.bounded_iff
tff(fact_1006_map__is__Nil__conv,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn(F,Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
tff(fact_1007_map__is__Nil__conv,axiom,
    ! [F: fun_in6898239422062238174n_assn,Xs: list_int] :
      ( ( map_in4427992030928829247n_assn(F,Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_int ) ) ).

% map_is_Nil_conv
tff(fact_1008_map__is__Nil__conv,axiom,
    ! [F: fun_na5138202758309126970n_assn,Xs: list_nat] :
      ( ( map_na2667955367175718043n_assn(F,Xs) = nil_Pr5671120429643327159n_assn )
    <=> ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
tff(fact_1009_map__is__Nil__conv,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,Xs: list_P8527749157015355191n_assn] :
      ( ( map_Pr7568062423562401049sn_int(F,Xs) = nil_int )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
tff(fact_1010_map__is__Nil__conv,axiom,
    ! [F: fun_int_int,Xs: list_int] :
      ( ( map_int_int(F,Xs) = nil_int )
    <=> ( Xs = nil_int ) ) ).

% map_is_Nil_conv
tff(fact_1011_map__is__Nil__conv,axiom,
    ! [F: fun_nat_int,Xs: list_nat] :
      ( ( map_nat_int(F,Xs) = nil_int )
    <=> ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
tff(fact_1012_map__is__Nil__conv,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,Xs: list_P8527749157015355191n_assn] :
      ( ( map_Pr7570552894071451325sn_nat(F,Xs) = nil_nat )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
tff(fact_1013_map__is__Nil__conv,axiom,
    ! [F: fun_int_nat,Xs: list_int] :
      ( ( map_int_nat(F,Xs) = nil_nat )
    <=> ( Xs = nil_int ) ) ).

% map_is_Nil_conv
tff(fact_1014_map__is__Nil__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = nil_nat )
    <=> ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
tff(fact_1015_map__is__Nil__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = nil_assn )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% map_is_Nil_conv
tff(fact_1016_Nil__is__map__conv,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn = map_Pr7925354932063753860n_assn(F,Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
tff(fact_1017_Nil__is__map__conv,axiom,
    ! [F: fun_in6898239422062238174n_assn,Xs: list_int] :
      ( ( nil_Pr5671120429643327159n_assn = map_in4427992030928829247n_assn(F,Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_map_conv
tff(fact_1018_Nil__is__map__conv,axiom,
    ! [F: fun_na5138202758309126970n_assn,Xs: list_nat] :
      ( ( nil_Pr5671120429643327159n_assn = map_na2667955367175718043n_assn(F,Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
tff(fact_1019_Nil__is__map__conv,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_int = map_Pr7568062423562401049sn_int(F,Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
tff(fact_1020_Nil__is__map__conv,axiom,
    ! [F: fun_int_int,Xs: list_int] :
      ( ( nil_int = map_int_int(F,Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_map_conv
tff(fact_1021_Nil__is__map__conv,axiom,
    ! [F: fun_nat_int,Xs: list_nat] :
      ( ( nil_int = map_nat_int(F,Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
tff(fact_1022_Nil__is__map__conv,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_nat = map_Pr7570552894071451325sn_nat(F,Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
tff(fact_1023_Nil__is__map__conv,axiom,
    ! [F: fun_int_nat,Xs: list_int] :
      ( ( nil_nat = map_int_nat(F,Xs) )
    <=> ( Xs = nil_int ) ) ).

% Nil_is_map_conv
tff(fact_1024_Nil__is__map__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] :
      ( ( nil_nat = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) )
    <=> ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
tff(fact_1025_Nil__is__map__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nil_assn = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_map_conv
tff(fact_1026_list_Omap__disc__iff,axiom,
    ! [F: fun_Pr5530265736160528387n_assn,A: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn(F,A) = nil_Pr5671120429643327159n_assn )
    <=> ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
tff(fact_1027_list_Omap__disc__iff,axiom,
    ! [F: fun_in6898239422062238174n_assn,A: list_int] :
      ( ( map_in4427992030928829247n_assn(F,A) = nil_Pr5671120429643327159n_assn )
    <=> ( A = nil_int ) ) ).

% list.map_disc_iff
tff(fact_1028_list_Omap__disc__iff,axiom,
    ! [F: fun_na5138202758309126970n_assn,A: list_nat] :
      ( ( map_na2667955367175718043n_assn(F,A) = nil_Pr5671120429643327159n_assn )
    <=> ( A = nil_nat ) ) ).

% list.map_disc_iff
tff(fact_1029_list_Omap__disc__iff,axiom,
    ! [F: fun_Pr3225707273362898288sn_int,A: list_P8527749157015355191n_assn] :
      ( ( map_Pr7568062423562401049sn_int(F,A) = nil_int )
    <=> ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
tff(fact_1030_list_Omap__disc__iff,axiom,
    ! [F: fun_int_int,A: list_int] :
      ( ( map_int_int(F,A) = nil_int )
    <=> ( A = nil_int ) ) ).

% list.map_disc_iff
tff(fact_1031_list_Omap__disc__iff,axiom,
    ! [F: fun_nat_int,A: list_nat] :
      ( ( map_nat_int(F,A) = nil_int )
    <=> ( A = nil_nat ) ) ).

% list.map_disc_iff
tff(fact_1032_list_Omap__disc__iff,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat,A: list_P8527749157015355191n_assn] :
      ( ( map_Pr7570552894071451325sn_nat(F,A) = nil_nat )
    <=> ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
tff(fact_1033_list_Omap__disc__iff,axiom,
    ! [F: fun_int_nat,A: list_int] :
      ( ( map_int_nat(F,A) = nil_nat )
    <=> ( A = nil_int ) ) ).

% list.map_disc_iff
tff(fact_1034_list_Omap__disc__iff,axiom,
    ! [F: fun_nat_nat,A: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),A) = nil_nat )
    <=> ( A = nil_nat ) ) ).

% list.map_disc_iff
tff(fact_1035_list_Omap__disc__iff,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,A: list_P8527749157015355191n_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),A) = nil_assn )
    <=> ( A = nil_Pr5671120429643327159n_assn ) ) ).

% list.map_disc_iff
tff(fact_1036_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2))
    <=> ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2)
        & aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B2) ) ) ).

% Int_subset_iff
tff(fact_1037_Int__subset__iff,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2))
    <=> ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2)
        & aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B2) ) ) ).

% Int_subset_iff
tff(fact_1038_nths__nil,axiom,
    ! [A2: set_nat] : nths_P5537322223065504949n_assn(nil_Pr5671120429643327159n_assn,A2) = nil_Pr5671120429643327159n_assn ).

% nths_nil
tff(fact_1039_nths__nil,axiom,
    ! [A2: set_nat] : nths_int(nil_int,A2) = nil_int ).

% nths_nil
tff(fact_1040_nths__nil,axiom,
    ! [A2: set_nat] : nths_nat(nil_nat,A2) = nil_nat ).

% nths_nil
tff(fact_1041_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) ) ).

% neg_0_le_iff_le
tff(fact_1042_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% neg_0_le_iff_le
tff(fact_1043_neg__0__le__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% neg_0_le_iff_le
tff(fact_1044_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A) ) ).

% neg_le_0_iff_le
tff(fact_1045_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).

% neg_le_0_iff_le
tff(fact_1046_neg__le__0__iff__le,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_le_0_iff_le
tff(fact_1047_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) ) ).

% less_eq_neg_nonpos
tff(fact_1048_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% less_eq_neg_nonpos
tff(fact_1049_less__eq__neg__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% less_eq_neg_nonpos
tff(fact_1050_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A) ) ).

% neg_less_eq_nonneg
tff(fact_1051_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),A)
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).

% neg_less_eq_nonneg
tff(fact_1052_neg__less__eq__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% neg_less_eq_nonneg
tff(fact_1053_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
tff(fact_1054_mult__cancel__right2,axiom,
    ! [A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
tff(fact_1055_mult__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
tff(fact_1056_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
tff(fact_1057_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
tff(fact_1058_mult__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
tff(fact_1059_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = C )
    <=> ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
tff(fact_1060_mult__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = C )
    <=> ( ( C = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
tff(fact_1061_mult__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = C )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
tff(fact_1062_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
    <=> ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
tff(fact_1063_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
    <=> ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
tff(fact_1064_mult__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
    <=> ( ( C = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
tff(fact_1065_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A2: set_int] :
      ( ~ finite_finite_int(A2)
     => ( linord2612477271533052124et_int(A2) = nil_int ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_1066_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A2: set_nat] :
      ( ~ finite_finite_nat(A2)
     => ( linord2614967742042102400et_nat(A2) = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
tff(fact_1067_zero__le,axiom,
    ! [X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X) ).

% zero_le
tff(fact_1068_nths__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,I: set_nat] : nths_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs),I) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),nths_nat(Xs,I)) ).

% nths_map
tff(fact_1069_nths__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,I: set_nat] : nths_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs),I) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),nths_P5537322223065504949n_assn(Xs,I)) ).

% nths_map
tff(fact_1070_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
    <=> ( X = zero_zero_nat ) ) ).

% zero_reorient
tff(fact_1071_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
    <=> ( X = zero_zero_int ) ) ).

% zero_reorient
tff(fact_1072_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
    <=> ( X = zero_zero_rat ) ) ).

% zero_reorient
tff(fact_1073_zero__reorient,axiom,
    ! [X: code_integer] :
      ( ( zero_z3403309356797280102nteger = X )
    <=> ( X = zero_z3403309356797280102nteger ) ) ).

% zero_reorient
tff(fact_1074_zero__reorient,axiom,
    ! [X: code_natural] :
      ( ( zero_z2226904508553997617atural = X )
    <=> ( X = zero_z2226904508553997617atural ) ) ).

% zero_reorient
tff(fact_1075_times__assn__raw_Ocases,axiom,
    ! [X: produc3112576180687729086et_nat] :
      ~ ! [P3: fun_Pr6460944699146269935_nat_o,Q3: fun_Pr6460944699146269935_nat_o,H3: heap_e7401611519738050253t_unit,As: set_nat] : X != produc7291855603632349880et_nat(P3,produc6017135503889078682et_nat(Q3,produc7507926704131184380et_nat(H3,As))) ).

% times_assn_raw.cases
tff(fact_1076_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1077_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1078_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1079_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1080_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1081_zero__le__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_le_mult_iff
tff(fact_1082_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) )
        | ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) ) ) ).

% zero_le_mult_iff
tff(fact_1083_zero__le__mult__iff,axiom,
    ! [A: int,B: 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,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) ) ) ).

% zero_le_mult_iff
tff(fact_1084_mult__nonneg__nonpos2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1085_mult__nonneg__nonpos2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),zero_z2226904508553997617atural) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1086_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1087_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),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,B),A)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1088_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,B),A)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1089_mult__nonpos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonpos_nonneg
tff(fact_1090_mult__nonpos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_nonpos_nonneg
tff(fact_1091_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1092_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_nonpos_nonneg
tff(fact_1093_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_nonpos_nonneg
tff(fact_1094_mult__nonneg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_nonneg_nonpos
tff(fact_1095_mult__nonneg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_nonneg_nonpos
tff(fact_1096_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1097_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% mult_nonneg_nonpos
tff(fact_1098_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)),zero_zero_int) ) ) ).

% mult_nonneg_nonpos
tff(fact_1099_mult__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1100_mult__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1101_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1102_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1103_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% mult_nonneg_nonneg
tff(fact_1104_split__mult__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) ) )
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ).

% split_mult_neg_le
tff(fact_1105_split__mult__neg__le,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
          & aa_Code_natural_o(ord_le1926595141338095240atural(B),zero_z2226904508553997617atural) )
        | ( aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural)
          & aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B) ) )
     => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural) ) ).

% split_mult_neg_le
tff(fact_1106_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) )
        | ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) )
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ).

% split_mult_neg_le
tff(fact_1107_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),zero_zero_nat) )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ).

% split_mult_neg_le
tff(fact_1108_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ).

% split_mult_neg_le
tff(fact_1109_mult__le__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1110_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) )
        | ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1111_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ) ) ).

% mult_le_0_iff
tff(fact_1112_mult__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono
tff(fact_1113_mult__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ) ).

% mult_right_mono
tff(fact_1114_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono
tff(fact_1115_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_right_mono
tff(fact_1116_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono
tff(fact_1117_mult__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1118_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1119_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_right_mono_neg
tff(fact_1120_mult__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono
tff(fact_1121_mult__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ) ) ).

% mult_left_mono
tff(fact_1122_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono
tff(fact_1123_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_left_mono
tff(fact_1124_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono
tff(fact_1125_mult__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1126_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1127_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)) ) ) ).

% mult_nonpos_nonpos
tff(fact_1128_mult__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1129_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1130_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_left_mono_neg
tff(fact_1131_split__mult__pos__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger) ) )
     => aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ).

% split_mult_pos_le
tff(fact_1132_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) )
        | ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) )
     => aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ).

% split_mult_pos_le
tff(fact_1133_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)) ) ).

% split_mult_pos_le
tff(fact_1134_zero__le__square,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)) ).

% zero_le_square
tff(fact_1135_zero__le__square,axiom,
    ! [A: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)) ).

% zero_le_square
tff(fact_1136_zero__le__square,axiom,
    ! [A: 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,A),A)) ).

% zero_le_square
tff(fact_1137_mult__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1138_mult__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(C),D2)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1139_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),D2)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1140_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1141_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono'
tff(fact_1142_mult__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1143_mult__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(C),D2)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
         => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1144_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),D2)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1145_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1146_mult__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_mono
tff(fact_1147_not__one__le__zero,axiom,
    ~ aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_le_zero
tff(fact_1148_not__one__le__zero,axiom,
    ~ aa_Code_natural_o(ord_le1926595141338095240atural(one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_le_zero
tff(fact_1149_not__one__le__zero,axiom,
    ~ aa_rat_o(ord_less_eq_rat(one_one_rat),zero_zero_rat) ).

% not_one_le_zero
tff(fact_1150_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_1151_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_1152_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),one_one_Code_integer) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1153_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),one_one_Code_natural) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1154_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    aa_rat_o(ord_less_eq_rat(zero_zero_rat),one_one_rat) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_1155_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_1156_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_1157_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one_class.zero_le_one
tff(fact_1158_zero__less__one__class_Ozero__le__one,axiom,
    aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one_class.zero_le_one
tff(fact_1159_zero__less__one__class_Ozero__le__one,axiom,
    aa_rat_o(ord_less_eq_rat(zero_zero_rat),one_one_rat) ).

% zero_less_one_class.zero_le_one
tff(fact_1160_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_1161_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_1162_mult__left__le,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),one_one_Code_integer)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),A) ) ) ).

% mult_left_le
tff(fact_1163_mult__left__le,axiom,
    ! [C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(C),one_one_Code_natural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),A) ) ) ).

% mult_left_le
tff(fact_1164_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),one_one_rat)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_1165_mult__left__le,axiom,
    ! [C: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),A) ) ) ).

% mult_left_le
tff(fact_1166_mult__left__le,axiom,
    ! [C: int,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),A) ) ) ).

% mult_left_le
tff(fact_1167_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),one_one_Code_integer)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ) ).

% mult_le_one
tff(fact_1168_mult__le__one,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),one_one_Code_natural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),one_one_Code_natural)
         => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),one_one_Code_natural) ) ) ) ).

% mult_le_one
tff(fact_1169_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
       => ( aa_rat_o(ord_less_eq_rat(B),one_one_rat)
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),one_one_rat) ) ) ) ).

% mult_le_one
tff(fact_1170_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),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,A),B)),one_one_nat) ) ) ) ).

% mult_le_one
tff(fact_1171_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)),one_one_int) ) ) ) ).

% mult_le_one
tff(fact_1172_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(Y),one_one_Code_integer)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_1173_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
       => ( aa_rat_o(ord_less_eq_rat(Y),one_one_rat)
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_1174_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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,X),Y)),X) ) ) ) ).

% mult_right_le_one_le
tff(fact_1175_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),Y)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(Y),one_one_Code_integer)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_1176_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
       => ( aa_rat_o(ord_less_eq_rat(Y),one_one_rat)
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_1177_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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),X)),X) ) ) ) ).

% mult_left_le_one_le
tff(fact_1178_le__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% le_minus_one_simps(1)
tff(fact_1179_le__minus__one__simps_I1_J,axiom,
    aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% le_minus_one_simps(1)
tff(fact_1180_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_1181_le__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(3)
tff(fact_1182_le__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(ord_less_eq_rat(zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(3)
tff(fact_1183_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_1184_list_Osimps_I8_J,axiom,
    ! [F: fun_Pr5530265736160528387n_assn] : map_Pr7925354932063753860n_assn(F,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% list.simps(8)
tff(fact_1185_list_Osimps_I8_J,axiom,
    ! [F: fun_Pr3225707273362898288sn_int] : map_Pr7568062423562401049sn_int(F,nil_Pr5671120429643327159n_assn) = nil_int ).

% list.simps(8)
tff(fact_1186_list_Osimps_I8_J,axiom,
    ! [F: fun_Pr7403558292872094996sn_nat] : map_Pr7570552894071451325sn_nat(F,nil_Pr5671120429643327159n_assn) = nil_nat ).

% list.simps(8)
tff(fact_1187_list_Osimps_I8_J,axiom,
    ! [F: fun_in6898239422062238174n_assn] : map_in4427992030928829247n_assn(F,nil_int) = nil_Pr5671120429643327159n_assn ).

% list.simps(8)
tff(fact_1188_list_Osimps_I8_J,axiom,
    ! [F: fun_int_int] : map_int_int(F,nil_int) = nil_int ).

% list.simps(8)
tff(fact_1189_list_Osimps_I8_J,axiom,
    ! [F: fun_int_nat] : map_int_nat(F,nil_int) = nil_nat ).

% list.simps(8)
tff(fact_1190_list_Osimps_I8_J,axiom,
    ! [F: fun_na5138202758309126970n_assn] : map_na2667955367175718043n_assn(F,nil_nat) = nil_Pr5671120429643327159n_assn ).

% list.simps(8)
tff(fact_1191_list_Osimps_I8_J,axiom,
    ! [F: fun_nat_int] : map_nat_int(F,nil_nat) = nil_int ).

% list.simps(8)
tff(fact_1192_list_Osimps_I8_J,axiom,
    ! [F: fun_nat_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),nil_nat) = nil_nat ).

% list.simps(8)
tff(fact_1193_list_Osimps_I8_J,axiom,
    ! [F: fun_Pr3816910165196019820n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),nil_Pr5671120429643327159n_assn) = nil_assn ).

% list.simps(8)
tff(fact_1194_le__numeral__extra_I4_J,axiom,
    aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),one_one_Code_integer) ).

% le_numeral_extra(4)
tff(fact_1195_le__numeral__extra_I4_J,axiom,
    aa_rat_o(ord_less_eq_rat(one_one_rat),one_one_rat) ).

% le_numeral_extra(4)
tff(fact_1196_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_1197_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_1198_pure__assn__raw_Ocases,axiom,
    ! [X: produc6197004810343482825et_nat] :
      ~ ! [B3: $o,H3: heap_e7401611519738050253t_unit,As: set_nat] : X != produc3762353314782720579et_nat((B3),produc7507926704131184380et_nat(H3,As)) ).

% pure_assn_raw.cases
tff(fact_1199_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_minus_iff
tff(fact_1200_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),uminus_uminus_rat(B))
    <=> aa_rat_o(ord_less_eq_rat(B),uminus_uminus_rat(A)) ) ).

% le_minus_iff
tff(fact_1201_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% le_minus_iff
tff(fact_1202_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_le_iff
tff(fact_1203_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),B)
    <=> aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),A) ) ).

% minus_le_iff
tff(fact_1204_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_le_iff
tff(fact_1205_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% le_imp_neg_le
tff(fact_1206_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),uminus_uminus_rat(A)) ) ).

% le_imp_neg_le
tff(fact_1207_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ) ).

% le_imp_neg_le
tff(fact_1208_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1209_inf_OcoboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1210_inf_OcoboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),C)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1211_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1212_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),C)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1213_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1214_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI2
tff(fact_1215_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1216_inf_OcoboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1217_inf_OcoboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1218_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1219_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),C)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1220_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1221_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.coboundedI1
tff(fact_1222_inf_Oabsorb__iff2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1223_inf_Oabsorb__iff2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1224_inf_Oabsorb__iff2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1225_inf_Oabsorb__iff2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1226_inf_Oabsorb__iff2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1227_inf_Oabsorb__iff2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1228_inf_Oabsorb__iff2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb_iff2
tff(fact_1229_inf_Oabsorb__iff1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1230_inf_Oabsorb__iff1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1231_inf_Oabsorb__iff1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1232_inf_Oabsorb__iff1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1233_inf_Oabsorb__iff1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1234_inf_Oabsorb__iff1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1235_inf_Oabsorb__iff1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb_iff1
tff(fact_1236_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_1237_inf_Ocobounded2,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),B) ).

% inf.cobounded2
tff(fact_1238_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),B) ).

% inf.cobounded2
tff(fact_1239_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_1240_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),B) ).

% inf.cobounded2
tff(fact_1241_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),B) ).

% inf.cobounded2
tff(fact_1242_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),B) ).

% inf.cobounded2
tff(fact_1243_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_1244_inf_Ocobounded1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),A) ).

% inf.cobounded1
tff(fact_1245_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),A) ).

% inf.cobounded1
tff(fact_1246_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_1247_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),A) ).

% inf.cobounded1
tff(fact_1248_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),A) ).

% inf.cobounded1
tff(fact_1249_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),A) ).

% inf.cobounded1
tff(fact_1250_inf_Oorder__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
    <=> ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_1251_inf_Oorder__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
    <=> ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.order_iff
tff(fact_1252_inf_Oorder__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.order_iff
tff(fact_1253_inf_Oorder__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_1254_inf_Oorder__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.order_iff
tff(fact_1255_inf_Oorder__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.order_iff
tff(fact_1256_inf_Oorder__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.order_iff
tff(fact_1257_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Z)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1258_inf__greatest,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Z)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1259_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Z)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1260_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Z)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1261_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),Y)
     => ( aa_rat_o(ord_less_eq_rat(X),Z)
       => aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1262_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Z)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1263_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Z)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z)) ) ) ).

% inf_greatest
tff(fact_1264_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_1265_inf_OboundedI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C)) ) ) ).

% inf.boundedI
tff(fact_1266_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C)) ) ) ).

% inf.boundedI
tff(fact_1267_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_1268_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(A),C)
       => aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C)) ) ) ).

% inf.boundedI
tff(fact_1269_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C)) ) ) ).

% inf.boundedI
tff(fact_1270_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C)) ) ) ).

% inf.boundedI
tff(fact_1271_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_1272_inf_OboundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C) ) ) ).

% inf.boundedE
tff(fact_1273_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C) ) ) ).

% inf.boundedE
tff(fact_1274_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_1275_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(ord_less_eq_rat(A),B)
         => ~ aa_rat_o(ord_less_eq_rat(A),C) ) ) ).

% inf.boundedE
tff(fact_1276_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C) ) ) ).

% inf.boundedE
tff(fact_1277_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C) ) ) ).

% inf.boundedE
tff(fact_1278_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,Y),X)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1279_inf__absorb2,axiom,
    ! [Y: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,Y),X)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1280_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1281_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1282_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(ord_less_eq_rat(Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1283_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1284_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = Y ) ) ).

% inf_absorb2
tff(fact_1285_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1286_inf__absorb1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1287_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1288_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1289_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1290_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1291_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% inf_absorb1
tff(fact_1292_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1293_inf_Oabsorb2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1294_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1295_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1296_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1297_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1298_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb2
tff(fact_1299_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1300_inf_Oabsorb1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1301_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1302_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1303_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1304_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1305_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb1
tff(fact_1306_le__iff__inf,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),Y)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1307_le__iff__inf,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),Y)
    <=> ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1308_le__iff__inf,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1309_le__iff__inf,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1310_le__iff__inf,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1311_le__iff__inf,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1312_le__iff__inf,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = X ) ) ).

% le_iff_inf
tff(fact_1313_inf__unique,axiom,
    ! [F: fun_se3560235952666129655at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
              ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Y2)
             => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),Z2)
               => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,Y2),Z2)) ) )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1314_inf__unique,axiom,
    ! [F: fun_fu959379683396315552_int_o,X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X2),Y2)),X2)
     => ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X2),Y2)),Y2)
       => ( ! [X2: fun_Pr3238374608701176162_int_o,Y2: fun_Pr3238374608701176162_int_o,Z2: fun_Pr3238374608701176162_int_o] :
              ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),Y2)
             => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),Z2)
               => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X2),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,Y2),Z2)) ) )
         => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1315_inf__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),X2)
     => ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2)),Y2)
       => ( ! [X2: assn,Y2: assn,Z2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Y2)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),Z2)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z2)) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1316_inf__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),X2)
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2)),Y2)
       => ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Y2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),Z2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1317_inf__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X2: rat,Y2: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2)),X2)
     => ( ! [X2: rat,Y2: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2)),Y2)
       => ( ! [X2: rat,Y2: rat,Z2: rat] :
              ( aa_rat_o(ord_less_eq_rat(X2),Y2)
             => ( aa_rat_o(ord_less_eq_rat(X2),Z2)
               => aa_rat_o(ord_less_eq_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z2)) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1318_inf__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: 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(F,X2),Y2)),X2)
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2)),Y2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Y2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),Z2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1319_inf__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),X2)
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,X2),Y2)),Y2)
       => ( ! [X2: int,Y2: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Y2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),Z2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,Y2),Z2)) ) )
         => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% inf_unique
tff(fact_1320_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B) ) ).

% inf.orderI
tff(fact_1321_inf_OorderI,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B) ) ).

% inf.orderI
tff(fact_1322_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B) ) ).

% inf.orderI
tff(fact_1323_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B) ) ).

% inf.orderI
tff(fact_1324_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
     => aa_rat_o(ord_less_eq_rat(A),B) ) ).

% inf.orderI
tff(fact_1325_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% inf.orderI
tff(fact_1326_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% inf.orderI
tff(fact_1327_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),B)
     => ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) ) ) ).

% inf.orderE
tff(fact_1328_inf_OorderE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),B)
     => ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) ) ) ).

% inf.orderE
tff(fact_1329_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% inf.orderE
tff(fact_1330_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) ) ) ).

% inf.orderE
tff(fact_1331_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) ) ) ).

% inf.orderE
tff(fact_1332_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) ) ) ).

% inf.orderE
tff(fact_1333_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) ) ) ).

% inf.orderE
tff(fact_1334_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI2
tff(fact_1335_le__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI2
tff(fact_1336_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI2
tff(fact_1337_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI2
tff(fact_1338_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),X)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI2
tff(fact_1339_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI2
tff(fact_1340_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI2
tff(fact_1341_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% le_infI1
tff(fact_1342_le__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% le_infI1
tff(fact_1343_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% le_infI1
tff(fact_1344_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% le_infI1
tff(fact_1345_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),X)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% le_infI1
tff(fact_1346_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% le_infI1
tff(fact_1347_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% le_infI1
tff(fact_1348_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),C)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),D2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_1349_inf__mono,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,D2: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,A),C)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,B),D2)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,C),D2)) ) ) ).

% inf_mono
tff(fact_1350_inf__mono,axiom,
    ! [A: assn,C: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,C),D2)) ) ) ).

% inf_mono
tff(fact_1351_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_1352_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),C)
     => ( aa_rat_o(ord_less_eq_rat(B),D2)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,C),D2)) ) ) ).

% inf_mono
tff(fact_1353_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,C),D2)) ) ) ).

% inf_mono
tff(fact_1354_inf__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),aa_int_int(aa_int_fun_int_int(inf_inf_int,C),D2)) ) ) ).

% inf_mono
tff(fact_1355_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)) ) ) ).

% le_infI
tff(fact_1356_le__infI,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
     => ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B)
       => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)) ) ) ).

% le_infI
tff(fact_1357_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)) ) ) ).

% le_infI
tff(fact_1358_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)) ) ) ).

% le_infI
tff(fact_1359_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),A)
     => ( aa_rat_o(ord_less_eq_rat(X),B)
       => aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)) ) ) ).

% le_infI
tff(fact_1360_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)) ) ) ).

% le_infI
tff(fact_1361_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)) ) ) ).

% le_infI
tff(fact_1362_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),A)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),B) ) ) ).

% le_infE
tff(fact_1363_le__infE,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),A)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),B) ) ) ).

% le_infE
tff(fact_1364_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B) ) ) ).

% le_infE
tff(fact_1365_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B) ) ) ).

% le_infE
tff(fact_1366_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B))
     => ~ ( aa_rat_o(ord_less_eq_rat(X),A)
         => ~ aa_rat_o(ord_less_eq_rat(X),B) ) ) ).

% le_infE
tff(fact_1367_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B) ) ) ).

% le_infE
tff(fact_1368_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B) ) ) ).

% le_infE
tff(fact_1369_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_le2
tff(fact_1370_inf__le2,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_le2
tff(fact_1371_inf__le2,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_le2
tff(fact_1372_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_le2
tff(fact_1373_inf__le2,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_le2
tff(fact_1374_inf__le2,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_le2
tff(fact_1375_inf__le2,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_le2
tff(fact_1376_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_le1
tff(fact_1377_inf__le1,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_le1
tff(fact_1378_inf__le1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_le1
tff(fact_1379_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_le1
tff(fact_1380_inf__le1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_le1
tff(fact_1381_inf__le1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_le1
tff(fact_1382_inf__le1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_le1
tff(fact_1383_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1384_inf__sup__ord_I1_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1385_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1386_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1387_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1388_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1389_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),X) ).

% inf_sup_ord(1)
tff(fact_1390_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1391_inf__sup__ord_I2_J,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1392_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1393_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1394_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1395_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1396_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),Y) ).

% inf_sup_ord(2)
tff(fact_1397_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1398_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1399_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(ord_less_eq_rat(Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1400_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1401_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(4)
tff(fact_1402_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1403_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1404_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1405_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1406_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% inf_sup_ord(3)
tff(fact_1407_le__supE,axiom,
    ! [A: assn,B: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X) ) ) ).

% le_supE
tff(fact_1408_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X) ) ) ).

% le_supE
tff(fact_1409_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X)
     => ~ ( aa_rat_o(ord_less_eq_rat(A),X)
         => ~ aa_rat_o(ord_less_eq_rat(B),X) ) ) ).

% le_supE
tff(fact_1410_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X) ) ) ).

% le_supE
tff(fact_1411_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X) ) ) ).

% le_supE
tff(fact_1412_le__supI,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),X)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),X)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),X) ) ) ).

% le_supI
tff(fact_1413_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_1414_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),X)
     => ( aa_rat_o(ord_less_eq_rat(B),X)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),X) ) ) ).

% le_supI
tff(fact_1415_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),X) ) ) ).

% le_supI
tff(fact_1416_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),X) ) ) ).

% le_supI
tff(fact_1417_sup__ge1,axiom,
    ! [X: assn,Y: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% sup_ge1
tff(fact_1418_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge1
tff(fact_1419_sup__ge1,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge1
tff(fact_1420_sup__ge1,axiom,
    ! [X: nat,Y: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge1
tff(fact_1421_sup__ge1,axiom,
    ! [X: int,Y: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge1
tff(fact_1422_sup__ge2,axiom,
    ! [Y: assn,X: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)) ).

% sup_ge2
tff(fact_1423_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)) ).

% sup_ge2
tff(fact_1424_sup__ge2,axiom,
    ! [Y: rat,X: rat] : aa_rat_o(ord_less_eq_rat(Y),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)) ).

% sup_ge2
tff(fact_1425_sup__ge2,axiom,
    ! [Y: nat,X: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)) ).

% sup_ge2
tff(fact_1426_sup__ge2,axiom,
    ! [Y: int,X: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)) ).

% sup_ge2
tff(fact_1427_le__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI1
tff(fact_1428_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI1
tff(fact_1429_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),A)
     => aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI1
tff(fact_1430_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI1
tff(fact_1431_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI1
tff(fact_1432_le__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% le_supI2
tff(fact_1433_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% le_supI2
tff(fact_1434_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),B)
     => aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% le_supI2
tff(fact_1435_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% le_supI2
tff(fact_1436_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% le_supI2
tff(fact_1437_sup_Omono,axiom,
    ! [C: assn,A: assn,D2: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,D2),B)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ) ).

% sup.mono
tff(fact_1438_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,D2),B)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1439_sup_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),A)
     => ( aa_rat_o(ord_less_eq_rat(D2),B)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D2)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ) ).

% sup.mono
tff(fact_1440_sup_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ) ).

% sup.mono
tff(fact_1441_sup_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ) ).

% sup.mono
tff(fact_1442_sup__mono,axiom,
    ! [A: assn,C: assn,B: assn,D2: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),C)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),D2)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,C),D2)) ) ) ).

% sup_mono
tff(fact_1443_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),C)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),D2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_1444_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),C)
     => ( aa_rat_o(ord_less_eq_rat(B),D2)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,C),D2)) ) ) ).

% sup_mono
tff(fact_1445_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),C)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,C),D2)) ) ) ).

% sup_mono
tff(fact_1446_sup__mono,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)),aa_int_int(aa_int_fun_int_int(sup_sup_int,C),D2)) ) ) ).

% sup_mono
tff(fact_1447_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z),X)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z)),X) ) ) ).

% sup_least
tff(fact_1448_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z),X)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_1449_sup__least,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(ord_less_eq_rat(Y),X)
     => ( aa_rat_o(ord_less_eq_rat(Z),X)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_1450_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z)),X) ) ) ).

% sup_least
tff(fact_1451_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z)),X) ) ) ).

% sup_least
tff(fact_1452_le__iff__sup,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1453_le__iff__sup,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1454_le__iff__sup,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),Y)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1455_le__iff__sup,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1456_le__iff__sup,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% le_iff_sup
tff(fact_1457_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.orderE
tff(fact_1458_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1459_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.orderE
tff(fact_1460_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.orderE
tff(fact_1461_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.orderE
tff(fact_1462_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A) ) ).

% sup.orderI
tff(fact_1463_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A) ) ).

% sup.orderI
tff(fact_1464_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
     => aa_rat_o(ord_less_eq_rat(B),A) ) ).

% sup.orderI
tff(fact_1465_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% sup.orderI
tff(fact_1466_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% sup.orderI
tff(fact_1467_sup__unique,axiom,
    ! [F: fun_as3983628580203722482n_assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
     => ( ! [X2: assn,Y2: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),aa_assn_assn(aa_ass954072293579886259n_assn(F,X2),Y2))
       => ( ! [X2: assn,Y2: assn,Z2: assn] :
              ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y2),X2)
             => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Z2),X2)
               => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(F,Y2),Z2)),X2) ) )
         => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1468_sup__unique,axiom,
    ! [F: fun_se1909708154429037836et_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
     => ( ! [X2: set_nat,Y2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X2),Y2))
       => ( ! [X2: set_nat,Y2: set_nat,Z2: set_nat] :
              ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y2),X2)
             => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Z2),X2)
               => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,Y2),Z2)),X2) ) )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1469_sup__unique,axiom,
    ! [F: fun_rat_fun_rat_rat,X: rat,Y: rat] :
      ( ! [X2: rat,Y2: rat] : aa_rat_o(ord_less_eq_rat(X2),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2))
     => ( ! [X2: rat,Y2: rat] : aa_rat_o(ord_less_eq_rat(Y2),aa_rat_rat(aa_rat_fun_rat_rat(F,X2),Y2))
       => ( ! [X2: rat,Y2: rat,Z2: rat] :
              ( aa_rat_o(ord_less_eq_rat(Y2),X2)
             => ( aa_rat_o(ord_less_eq_rat(Z2),X2)
               => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(F,Y2),Z2)),X2) ) )
         => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1470_sup__unique,axiom,
    ! [F: fun_nat_fun_nat_nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
     => ( ! [X2: nat,Y2: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),aa_nat_nat(aa_nat_fun_nat_nat(F,X2),Y2))
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y2),X2)
             => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z2),X2)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(F,Y2),Z2)),X2) ) )
         => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1471_sup__unique,axiom,
    ! [F: fun_int_fun_int_int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
     => ( ! [X2: int,Y2: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),aa_int_int(aa_int_fun_int_int(F,X2),Y2))
       => ( ! [X2: int,Y2: int,Z2: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y2),X2)
             => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z2),X2)
               => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(F,Y2),Z2)),X2) ) )
         => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = aa_int_int(aa_int_fun_int_int(F,X),Y) ) ) ) ) ).

% sup_unique
tff(fact_1472_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1473_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1474_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1475_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1476_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb1
tff(fact_1477_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1478_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1479_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1480_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1481_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb2
tff(fact_1482_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,Y),X)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1483_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,Y),X)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1484_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( aa_rat_o(ord_less_eq_rat(Y),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1485_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Y),X)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1486_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),X)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = X ) ) ).

% sup_absorb1
tff(fact_1487_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),Y)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1488_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),Y)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1489_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),Y)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1490_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1491_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Y)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y) = Y ) ) ).

% sup_absorb2
tff(fact_1492_sup_OboundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A) ) ) ).

% sup.boundedE
tff(fact_1493_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_1494_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(ord_less_eq_rat(B),A)
         => ~ aa_rat_o(ord_less_eq_rat(C),A) ) ) ).

% sup.boundedE
tff(fact_1495_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% sup.boundedE
tff(fact_1496_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% sup.boundedE
tff(fact_1497_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
     => ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
       => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_1498_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_1499_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_o(ord_less_eq_rat(C),A)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_1500_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_1501_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A) ) ) ).

% sup.boundedI
tff(fact_1502_sup_Oorder__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) ) ) ).

% sup.order_iff
tff(fact_1503_sup_Oorder__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1504_sup_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) ) ) ).

% sup.order_iff
tff(fact_1505_sup_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) ) ) ).

% sup.order_iff
tff(fact_1506_sup_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) ) ) ).

% sup.order_iff
tff(fact_1507_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded1
tff(fact_1508_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded1
tff(fact_1509_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded1
tff(fact_1510_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded1
tff(fact_1511_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded1
tff(fact_1512_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ).

% sup.cobounded2
tff(fact_1513_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ).

% sup.cobounded2
tff(fact_1514_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ).

% sup.cobounded2
tff(fact_1515_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ).

% sup.cobounded2
tff(fact_1516_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ).

% sup.cobounded2
tff(fact_1517_sup_Oabsorb__iff1,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,B),A)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1518_sup_Oabsorb__iff1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),A)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1519_sup_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1520_sup_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1521_sup_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb_iff1
tff(fact_1522_sup_Oabsorb__iff2,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1523_sup_Oabsorb__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),B)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1524_sup_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1525_sup_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1526_sup_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb_iff2
tff(fact_1527_sup_OcoboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI1
tff(fact_1528_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1529_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),A)
     => aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1530_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI1
tff(fact_1531_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI1
tff(fact_1532_sup_OcoboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.coboundedI2
tff(fact_1533_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1534_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),B)
     => aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1535_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.coboundedI2
tff(fact_1536_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.coboundedI2
tff(fact_1537_Int__Collect__mono,axiom,
    ! [A2: set_int,B2: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ord_less_eq_set_int(A2,B2)
     => ( ! [X2: int] :
            ( member_int2(X2,A2)
           => ( aa_int_o(Pa,X2)
             => aa_int_o(Qa,X2) ) )
       => ord_less_eq_set_int(aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),collect_int(Pa)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),collect_int(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1538_Int__Collect__mono,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o,Pa: fun_fu5135969500543783852at_o_o,Qa: fun_fu5135969500543783852at_o_o] :
      ( ord_le5394272759557949423_nat_o(A2,B2)
     => ( ! [X2: fun_Pr6460944699146269935_nat_o] :
            ( member7544390268018397336_nat_o(X2,A2)
           => ( aa_fun1675921005893230629at_o_o(Pa,X2)
             => aa_fun1675921005893230629at_o_o(Qa,X2) ) )
       => ord_le5394272759557949423_nat_o(inf_in3376754634429691197_nat_o(A2,collec6407654713642994138_nat_o(Pa)),inf_in3376754634429691197_nat_o(B2,collec6407654713642994138_nat_o(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1539_Int__Collect__mono,axiom,
    ! [A2: set_list_nat,B2: set_list_nat,Pa: fun_list_nat_o,Qa: fun_list_nat_o] :
      ( ord_le6045566169113846134st_nat(A2,B2)
     => ( ! [X2: list_nat] :
            ( member_list_nat(X2,A2)
           => ( aa_list_nat_o(Pa,X2)
             => aa_list_nat_o(Qa,X2) ) )
       => ord_le6045566169113846134st_nat(inf_inf_set_list_nat(A2,collect_list_nat(Pa)),inf_inf_set_list_nat(B2,collect_list_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1540_Int__Collect__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,Pa: fun_Pr4796943543331071514_nat_o,Qa: fun_Pr4796943543331071514_nat_o] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( ! [X2: product_prod_nat_nat] :
            ( member8440522571783428010at_nat(X2,A2)
           => ( aa_Pro6150637604445964947_nat_o(Pa,X2)
             => aa_Pro6150637604445964947_nat_o(Qa,X2) ) )
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),collec3392354462482085612at_nat(Pa))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),collec3392354462482085612at_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1541_Int__Collect__mono,axiom,
    ! [A2: set_nat,B2: set_nat,Pa: fun_nat_o,Qa: fun_nat_o] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( ! [X2: nat] :
            ( member_nat2(X2,A2)
           => ( aa_nat_o(Pa,X2)
             => aa_nat_o(Qa,X2) ) )
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),collect_nat(Pa))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),collect_nat(Qa))) ) ) ).

% Int_Collect_mono
tff(fact_1542_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),B2)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_1543_Int__greatest,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),B2)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_greatest
tff(fact_1544_Int__absorb2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),B2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_1545_Int__absorb2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),B2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = A2 ) ) ).

% Int_absorb2
tff(fact_1546_Int__absorb1,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_1547_Int__absorb1,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = B2 ) ) ).

% Int_absorb1
tff(fact_1548_Int__lower2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_1549_Int__lower2,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),B2) ).

% Int_lower2
tff(fact_1550_Int__lower1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_1551_Int__lower1,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),A2) ).

% Int_lower1
tff(fact_1552_Int__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A2),C2)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B2),D)
       => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),D)) ) ) ).

% Int_mono
tff(fact_1553_Int__mono,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat,D: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A2),C2)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B2),D)
       => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),D)) ) ) ).

% Int_mono
tff(fact_1554_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,S),S2)
     => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S2) )
       => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),S) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_1555_inter__eq__subsetI,axiom,
    ! [S: set_nat,S2: set_nat,A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,S),S2)
     => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S2) )
       => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),S) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),S) ) ) ) ).

% inter_eq_subsetI
tff(fact_1556_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1557_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1558_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),C) = aa_int_int(aa_int_fun_int_int(times_times_int,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1559_mult__right__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1560_mult__right__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C) )
      <=> ( A = B ) ) ) ).

% mult_right_cancel
tff(fact_1561_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1562_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1563_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,C),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1564_mult__left__cancel,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1565_mult__left__cancel,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B) )
      <=> ( A = B ) ) ) ).

% mult_left_cancel
tff(fact_1566_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat ) ) ) ).

% no_zero_divisors
tff(fact_1567_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat ) ) ) ).

% no_zero_divisors
tff(fact_1568_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int ) ) ) ).

% no_zero_divisors
tff(fact_1569_no__zero__divisors,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( B != zero_z3403309356797280102nteger )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger ) ) ) ).

% no_zero_divisors
tff(fact_1570_no__zero__divisors,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( B != zero_z2226904508553997617atural )
       => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) != zero_z2226904508553997617atural ) ) ) ).

% no_zero_divisors
tff(fact_1571_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
tff(fact_1572_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
tff(fact_1573_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
tff(fact_1574_divisors__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = zero_z3403309356797280102nteger )
     => ( ( A = zero_z3403309356797280102nteger )
        | ( B = zero_z3403309356797280102nteger ) ) ) ).

% divisors_zero
tff(fact_1575_divisors__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = zero_z2226904508553997617atural )
     => ( ( A = zero_z2226904508553997617atural )
        | ( B = zero_z2226904508553997617atural ) ) ) ).

% divisors_zero
tff(fact_1576_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B) != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
tff(fact_1577_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
tff(fact_1578_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
tff(fact_1579_mult__not__zero,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) != zero_z3403309356797280102nteger )
     => ( ( A != zero_z3403309356797280102nteger )
        & ( B != zero_z3403309356797280102nteger ) ) ) ).

% mult_not_zero
tff(fact_1580_mult__not__zero,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) != zero_z2226904508553997617atural )
     => ( ( A != zero_z2226904508553997617atural )
        & ( B != zero_z2226904508553997617atural ) ) ) ).

% mult_not_zero
tff(fact_1581_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
tff(fact_1582_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
tff(fact_1583_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
tff(fact_1584_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
tff(fact_1585_zero__neq__one,axiom,
    zero_z2226904508553997617atural != one_one_Code_natural ).

% zero_neq_one
tff(fact_1586_rotate1__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : rotate1_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),rotate1_nat(Xs)) ).

% rotate1_map
tff(fact_1587_rotate1__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : rotate1_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),rotate328796349445179396n_assn(Xs)) ).

% rotate1_map
tff(fact_1588_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( linord2614967742042102400et_nat(A2) = linord2614967742042102400et_nat(B2) )
     => ( finite_finite_nat(A2)
       => ( finite_finite_nat(B2)
         => ( A2 = B2 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
tff(fact_1589_less__eq__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
    <=> ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) ) ) ).

% less_eq_assn_def
tff(fact_1590_relH__subset,axiom,
    ! [Bs: set_nat,H2: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( relH(Bs,H2,H5)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As2),Bs)
       => relH(As2,H2,H5) ) ) ).

% relH_subset
tff(fact_1591_le__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% le_minus_one_simps(2)
tff(fact_1592_le__minus__one__simps_I2_J,axiom,
    aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% le_minus_one_simps(2)
tff(fact_1593_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_1594_le__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% le_minus_one_simps(4)
tff(fact_1595_le__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(ord_less_eq_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% le_minus_one_simps(4)
tff(fact_1596_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_1597_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_1598_distrib__inf__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Z))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,Y),Z))) ).

% distrib_inf_le
tff(fact_1599_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Y),Z))) ).

% distrib_inf_le
tff(fact_1600_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_1601_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,Y),Z))) ).

% distrib_inf_le
tff(fact_1602_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,Y),Z))) ).

% distrib_inf_le
tff(fact_1603_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Y)),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,Y),Z))) ).

% distrib_inf_le
tff(fact_1604_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Y),Z))),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Y)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,X),Z))) ).

% distrib_sup_le
tff(fact_1605_distrib__sup__le,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o,Z: fun_Pr3238374608701176162_int_o] : aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,Y),Z))),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Y)),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(sup_su4047618219528032398_int_o,X),Z))) ).

% distrib_sup_le
tff(fact_1606_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Y),Z))),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Y)),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,X),Z))) ).

% distrib_sup_le
tff(fact_1607_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Y),Z))),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Y)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,X),Z))) ).

% distrib_sup_le
tff(fact_1608_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,Y),Z))),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,X),Z))) ).

% distrib_sup_le
tff(fact_1609_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,Y),Z))),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,X),Z))) ).

% distrib_sup_le
tff(fact_1610_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),aa_int_int(aa_int_fun_int_int(inf_inf_int,Y),Z))),aa_int_int(aa_int_fun_int_int(inf_inf_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Y)),aa_int_int(aa_int_fun_int_int(sup_sup_int,X),Z))) ).

% distrib_sup_le
tff(fact_1611_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,A),A3)
     => ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,B),B4)
       => ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A3),B4) = bot_bo2099793752762293965at_nat )
         => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
tff(fact_1612_disjoint__mono,axiom,
    ! [A: set_int,A3: set_int,B: set_int,B4: set_int] :
      ( ord_less_eq_set_int(A,A3)
     => ( ord_less_eq_set_int(B,B4)
       => ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A3),B4) = bot_bot_set_int )
         => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A),B) = bot_bot_set_int ) ) ) ) ).

% disjoint_mono
tff(fact_1613_disjoint__mono,axiom,
    ! [A: set_nat,A3: set_nat,B: set_nat,B4: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,A),A3)
     => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,B),B4)
       => ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A3),B4) = bot_bot_set_nat )
         => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
tff(fact_1614_Un__Int__assoc__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_1615_Un__Int__assoc__eq,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,C2),A2) ) ).

% Un_Int_assoc_eq
tff(fact_1616_zero__neq__neg__one,axiom,
    zero_zero_int != aa_int_int(uminus_uminus_int,one_one_int) ).

% zero_neq_neg_one
tff(fact_1617_zero__neq__neg__one,axiom,
    zero_zero_rat != uminus_uminus_rat(one_one_rat) ).

% zero_neq_neg_one
tff(fact_1618_zero__neq__neg__one,axiom,
    zero_z3403309356797280102nteger != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% zero_neq_neg_one
tff(fact_1619_in__range__subset,axiom,
    ! [As2: set_nat,As3: set_nat,H2: heap_e7401611519738050253t_unit] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,As2),As3)
     => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As3))
       => aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2)) ) ) ).

% in_range_subset
tff(fact_1620_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),Y) = bot_bo2099793752762293965at_nat )
    <=> aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le3146513528884898305at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ) ).

% inf_shunt
tff(fact_1621_inf__shunt,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] :
      ( ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),Y) = bot_bo3516217578726220278_int_o )
    <=> aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le14110778710936002_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ) ).

% inf_shunt
tff(fact_1622_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,X),Y) = bot_bot_set_int )
    <=> ord_less_eq_set_int(X,aa_set_int_set_int(uminus1532241313380277803et_int,Y)) ) ).

% inf_shunt
tff(fact_1623_inf__shunt,axiom,
    ! [X: assn,Y: assn] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),Y) = bot_bot_assn )
    <=> aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ) ).

% inf_shunt
tff(fact_1624_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),Y) = bot_bot_set_nat )
    <=> aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ) ).

% inf_shunt
tff(fact_1625_finite__Int,axiom,
    ! [Fa: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat(Fa)
        | finite6177210948735845034at_nat(G2) )
     => finite6177210948735845034at_nat(aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,Fa),G2)) ) ).

% finite_Int
tff(fact_1626_finite__Int,axiom,
    ! [Fa: set_nat,G2: set_nat] :
      ( ( finite_finite_nat(Fa)
        | finite_finite_nat(G2) )
     => finite_finite_nat(aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,Fa),G2)) ) ).

% finite_Int
tff(fact_1627_wand__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As))))
               => ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                  & ! [H4: heap_e7401611519738050253t_unit,As4: set_nat] :
                      ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As4) = bot_bot_set_nat )
                        & relH(As,H3,H4)
                        & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H4,As))
                        & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H4,As4)) )
                     => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H4,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As4))) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
tff(fact_1628_wand__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As))))
               => ~ ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                    & ! [H6: heap_e7401611519738050253t_unit,As5: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As5) = bot_bot_set_nat )
                          & relH(As,H3,H6)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H6,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H6,As5)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H6,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As5))) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
tff(fact_1629_wand__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(wand_raw(X,Xa),Xb)
      <=> (Y) )
     => ( accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( ( (Y)
                <=> ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H3,As))
                    & ! [H7: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As),As6) = bot_bot_set_nat )
                          & relH(As,H3,H7)
                          & aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H7,As))
                          & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H7,As6)) )
                       => aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H7,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As),As6))) ) ) )
               => ~ accp_P1394777952347926645et_nat(wand_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As)))) ) ) ) ) ).

% wand_raw.pelims(1)
tff(fact_1630_mod__star__trueE_H,axiom,
    ! [Pa: assn,H2: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Pa),top_top_assn)),H2)
     => ~ ! [H4: produc3658429121746597890et_nat] :
            ( ( aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H4) = aa_Pro1219312880135979149t_unit(produc1824681642469235216et_nat,H2) )
           => ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_eq_set_nat,aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H4)),aa_Pro3416150927818442182et_nat(produc8586169260539613262et_nat,H2))
             => ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H4) ) ) ) ).

% mod_star_trueE'
tff(fact_1631_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
tff(fact_1632_map__eq__map__tailrec,axiom,
    map_Pr8991440229025900053n_assn = map_ta2194206859901583947n_assn ).

% map_eq_map_tailrec
tff(fact_1633_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
      <=> (Y) )
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( ( (Y)
                <=> ? [As1: set_nat,As22: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As1),As22) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As1),As22) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As1))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As22)) ) )
               => ~ accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As)))) ) ) ) ) ).

% times_assn_raw.pelims(1)
tff(fact_1634_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As))))
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As13),As24) )
                      & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As13),As24) = bot_bot_set_nat )
                      & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As13))
                      & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As24)) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
tff(fact_1635_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xa: fun_Pr6460944699146269935_nat_o,Xb: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(times_assn_raw(X,Xa),Xb)
     => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,Xb)))
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb = produc7507926704131184380et_nat(H3,As) )
             => ( accp_P1394777952347926645et_nat(times_assn_raw_rel,produc7291855603632349880et_nat(X,produc6017135503889078682et_nat(Xa,produc7507926704131184380et_nat(H3,As))))
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,As12),As23) )
                    & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,As12),As23) = bot_bot_set_nat )
                    & aa_Pro653082453696054616_nat_o(X,produc7507926704131184380et_nat(H3,As12))
                    & aa_Pro653082453696054616_nat_o(Xa,produc7507926704131184380et_nat(H3,As23)) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
tff(fact_1636_mult__le__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),B) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1637_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(one_one_rat),B) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(B),one_one_rat) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1638_mult__le__cancel__left1,axiom,
    ! [C: int,B: 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),B))
    <=> ( ( 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),B) )
        & ( 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,B),one_one_int) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1639_list_Oinject,axiom,
    ! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
      ( ( cons_int(X21,X22) = cons_int(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_1640_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( cons_nat(X21,X22) = cons_nat(Y21,Y22) )
    <=> ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
tff(fact_1641_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_1642_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% neg_less_iff_less
tff(fact_1643_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_rat(A),B) ) ).

% neg_less_iff_less
tff(fact_1644_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ).

% neg_less_iff_less
tff(fact_1645_list__ex__simps_I1_J,axiom,
    ! [Pa: fun_int_o,X: int,Xs: list_int] :
      ( aa_list_int_o(list_ex_int(Pa),cons_int(X,Xs))
    <=> ( aa_int_o(Pa,X)
        | aa_list_int_o(list_ex_int(Pa),Xs) ) ) ).

% list_ex_simps(1)
tff(fact_1646_list__ex__simps_I1_J,axiom,
    ! [Pa: fun_nat_o,X: nat,Xs: list_nat] :
      ( aa_list_nat_o(list_ex_nat(Pa),cons_nat(X,Xs))
    <=> ( aa_nat_o(Pa,X)
        | aa_list_nat_o(list_ex_nat(Pa),Xs) ) ) ).

% list_ex_simps(1)
tff(fact_1647_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F: fun_nat_nat,Fs: list_fun_nat_nat] :
      ( member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(cons_fun_nat_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
        | ( ( aa_nat_nat(F,X) = aa_nat_nat(F,Y) )
          & member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_1648_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F: fun_int_nat,Fs: list_fun_int_nat] :
      ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F,Fs)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,Y))
        | ( ( aa_int_nat(F,X) = aa_int_nat(F,Y) )
          & member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(Fs)) ) ) ) ).

% in_measures(2)
tff(fact_1649_neg__less__0__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_0_iff_less
tff(fact_1650_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).

% neg_less_0_iff_less
tff(fact_1651_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A) ) ).

% neg_less_0_iff_less
tff(fact_1652_neg__0__less__iff__less,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% neg_0_less_iff_less
tff(fact_1653_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% neg_0_less_iff_less
tff(fact_1654_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% neg_0_less_iff_less
tff(fact_1655_neg__less__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% neg_less_pos
tff(fact_1656_neg__less__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(A)),A)
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).

% neg_less_pos
tff(fact_1657_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A) ) ).

% neg_less_pos
tff(fact_1658_less__neg__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% less_neg_neg
tff(fact_1659_less__neg__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(A),uminus_uminus_rat(A))
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% less_neg_neg
tff(fact_1660_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% less_neg_neg
tff(fact_1661_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: nat,Ys: list_nat] : zip_int_nat(cons_int(X,Xs),cons_nat(Y,Ys)) = cons_P7512249878480867347nt_nat(product_Pair_int_nat(X,Y),zip_int_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1662_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: int,Ys: list_int] : zip_nat_int(cons_nat(X,Xs),cons_int(Y,Ys)) = cons_P2335045147070616083at_int(product_Pair_nat_int(X,Y),zip_nat_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1663_zip__Cons__Cons,axiom,
    ! [X: heap_e7401611519738050253t_unit,Xs: list_h2745205591237434579t_unit,Y: set_nat,Ys: list_set_nat] : zip_he6711929980790699996et_nat(cons_h4161061858534515843t_unit(X,Xs),cons_set_nat(Y,Ys)) = cons_P2869003950965171916et_nat(produc7507926704131184380et_nat(X,Y),zip_he6711929980790699996et_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1664_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] : zip_nat_nat(cons_nat(X,Xs),cons_nat(Y,Ys)) = cons_P6512896166579812791at_nat(product_Pair_nat_nat(X,Y),zip_nat_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1665_zip__Cons__Cons,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] : zip_int_int(cons_int(X,Xs),cons_int(Y,Ys)) = cons_P3334398858971670639nt_int(product_Pair_int_int(X,Y),zip_int_int(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1666_zip__Cons__Cons,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xs: list_f5222602229612168821_nat_o,Y: produc3658429121746597890et_nat,Ys: list_P9062070895058802706et_nat] : zip_fu1498620252286093946et_nat(cons_f8040665141538986917_nat_o(X,Xs),cons_P2869003950965171916et_nat(Y,Ys)) = cons_P8169920869477312234et_nat(produc6017135503889078682et_nat(X,Y),zip_fu1498620252286093946et_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1667_zip__Cons__Cons,axiom,
    ! [X: fun_Pr6460944699146269935_nat_o,Xs: list_f5222602229612168821_nat_o,Y: produc8799533983703829792et_nat,Ys: list_P8836981409019564976et_nat] : zip_fu1756369540273021848et_nat(cons_f8040665141538986917_nat_o(X,Xs),cons_P8169920869477312234et_nat(Y,Ys)) = cons_P5637129515692979848et_nat(produc7291855603632349880et_nat(X,Y),zip_fu1756369540273021848et_nat(Xs,Ys)) ).

% zip_Cons_Cons
tff(fact_1668_zip__map__fst__snd,axiom,
    ! [Zs: list_P9062070895058802706et_nat] : zip_he6711929980790699996et_nat(map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,Zs),map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_1669_zip__map__fst__snd,axiom,
    ! [Zs: list_P5707943133018811711nt_int] : zip_int_int(map_Pr6494458893431244577nt_int(product_fst_int_int,Zs),map_Pr6494458893431244577nt_int(product_snd_int_int,Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_1670_zip__map__fst__snd,axiom,
    ! [Zs: list_P8527749157015355191n_assn] : zip_assn_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),Zs),aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),Zs)) = Zs ).

% zip_map_fst_snd
tff(fact_1671_nths__singleton,axiom,
    ! [X: produc6575502325842934193n_assn,A2: set_nat] :
      nths_P5537322223065504949n_assn(cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn),A2) = $ite(member_nat2(zero_zero_nat,A2),cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn),nil_Pr5671120429643327159n_assn) ).

% nths_singleton
tff(fact_1672_nths__singleton,axiom,
    ! [X: int,A2: set_nat] :
      nths_int(cons_int(X,nil_int),A2) = $ite(member_nat2(zero_zero_nat,A2),cons_int(X,nil_int),nil_int) ).

% nths_singleton
tff(fact_1673_nths__singleton,axiom,
    ! [X: nat,A2: set_nat] :
      nths_nat(cons_nat(X,nil_nat),A2) = $ite(member_nat2(zero_zero_nat,A2),cons_nat(X,nil_nat),nil_nat) ).

% nths_singleton
tff(fact_1674_less__assn__def,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( aa_assn_o(aa_assn_fun_assn_o(ord_less_eq_assn,A),B)
        & ( A != B ) ) ) ).

% less_assn_def
tff(fact_1675_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] : cons_int(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_1676_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] : cons_nat(X,Xs) != Xs ).

% not_Cons_self2
tff(fact_1677_transpose_Ocases,axiom,
    ! [X: list_l6351802567095793725n_assn] :
      ( ( X != nil_li5476096274760905021n_assn )
     => ( ! [Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,Xss)
       => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Xss: list_l6351802567095793725n_assn] : X != cons_l2423627976422276333n_assn(cons_P2971678138204555879n_assn(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_1678_transpose_Ocases,axiom,
    ! [X: list_list_int] :
      ( ( X != nil_list_int )
     => ( ! [Xss: list_list_int] : X != cons_list_int(nil_int,Xss)
       => ~ ! [X2: int,Xs2: list_int,Xss: list_list_int] : X != cons_list_int(cons_int(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_1679_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss: list_list_nat] : X != cons_list_nat(nil_nat,Xss)
       => ~ ! [X2: nat,Xs2: list_nat,Xss: list_list_nat] : X != cons_list_nat(cons_nat(X2,Xs2),Xss) ) ) ).

% transpose.cases
tff(fact_1680_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_nat,Xy: product_prod_int_nat,Xys: list_P8198026277950538467nt_nat] :
      ( ( zip_int_nat(Xs,Ys) = cons_P7512249878480867347nt_nat(Xy,Xys) )
     => ~ ! [X2: int,Xs3: list_int] :
            ( ( Xs = cons_int(X2,Xs3) )
           => ! [Y2: nat,Ys2: list_nat] :
                ( ( Ys = cons_nat(Y2,Ys2) )
               => ( ( Xy = product_Pair_int_nat(X2,Y2) )
                 => ( Xys != zip_int_nat(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1681_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_int,Xy: product_prod_nat_int,Xys: list_P3521021558325789923at_int] :
      ( ( zip_nat_int(Xs,Ys) = cons_P2335045147070616083at_int(Xy,Xys) )
     => ~ ! [X2: nat,Xs3: list_nat] :
            ( ( Xs = cons_nat(X2,Xs3) )
           => ! [Y2: int,Ys2: list_int] :
                ( ( Ys = cons_int(Y2,Ys2) )
               => ( ( Xy = product_Pair_nat_int(X2,Y2) )
                 => ( Xys != zip_nat_int(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1682_zip__eq__ConsE,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat,Xy: produc3658429121746597890et_nat,Xys: list_P9062070895058802706et_nat] :
      ( ( zip_he6711929980790699996et_nat(Xs,Ys) = cons_P2869003950965171916et_nat(Xy,Xys) )
     => ~ ! [X2: heap_e7401611519738050253t_unit,Xs3: list_h2745205591237434579t_unit] :
            ( ( Xs = cons_h4161061858534515843t_unit(X2,Xs3) )
           => ! [Y2: set_nat,Ys2: list_set_nat] :
                ( ( Ys = cons_set_nat(Y2,Ys2) )
               => ( ( Xy = produc7507926704131184380et_nat(X2,Y2) )
                 => ( Xys != zip_he6711929980790699996et_nat(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1683_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( zip_nat_nat(Xs,Ys) = cons_P6512896166579812791at_nat(Xy,Xys) )
     => ~ ! [X2: nat,Xs3: list_nat] :
            ( ( Xs = cons_nat(X2,Xs3) )
           => ! [Y2: nat,Ys2: list_nat] :
                ( ( Ys = cons_nat(Y2,Ys2) )
               => ( ( Xy = product_Pair_nat_nat(X2,Y2) )
                 => ( Xys != zip_nat_nat(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1684_zip__eq__ConsE,axiom,
    ! [Xs: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( zip_int_int(Xs,Ys) = cons_P3334398858971670639nt_int(Xy,Xys) )
     => ~ ! [X2: int,Xs3: list_int] :
            ( ( Xs = cons_int(X2,Xs3) )
           => ! [Y2: int,Ys2: list_int] :
                ( ( Ys = cons_int(Y2,Ys2) )
               => ( ( Xy = product_Pair_int_int(X2,Y2) )
                 => ( Xys != zip_int_int(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1685_zip__eq__ConsE,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,Ys: list_P9062070895058802706et_nat,Xy: produc8799533983703829792et_nat,Xys: list_P8836981409019564976et_nat] :
      ( ( zip_fu1498620252286093946et_nat(Xs,Ys) = cons_P8169920869477312234et_nat(Xy,Xys) )
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Xs3: list_f5222602229612168821_nat_o] :
            ( ( Xs = cons_f8040665141538986917_nat_o(X2,Xs3) )
           => ! [Y2: produc3658429121746597890et_nat,Ys2: list_P9062070895058802706et_nat] :
                ( ( Ys = cons_P2869003950965171916et_nat(Y2,Ys2) )
               => ( ( Xy = produc6017135503889078682et_nat(X2,Y2) )
                 => ( Xys != zip_fu1498620252286093946et_nat(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1686_zip__eq__ConsE,axiom,
    ! [Xs: list_f5222602229612168821_nat_o,Ys: list_P8836981409019564976et_nat,Xy: produc3112576180687729086et_nat,Xys: list_P4783563340545138382et_nat] :
      ( ( zip_fu1756369540273021848et_nat(Xs,Ys) = cons_P5637129515692979848et_nat(Xy,Xys) )
     => ~ ! [X2: fun_Pr6460944699146269935_nat_o,Xs3: list_f5222602229612168821_nat_o] :
            ( ( Xs = cons_f8040665141538986917_nat_o(X2,Xs3) )
           => ! [Y2: produc8799533983703829792et_nat,Ys2: list_P8836981409019564976et_nat] :
                ( ( Ys = cons_P8169920869477312234et_nat(Y2,Ys2) )
               => ( ( Xy = produc7291855603632349880et_nat(X2,Y2) )
                 => ( Xys != zip_fu1756369540273021848et_nat(Xs3,Ys2) ) ) ) ) ) ).

% zip_eq_ConsE
tff(fact_1687_pair__list__eqI,axiom,
    ! [Xs: list_P9062070895058802706et_nat,Ys: list_P9062070895058802706et_nat] :
      ( ( map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,Xs) = map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,Ys) )
     => ( ( map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,Xs) = map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_1688_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_1689_pair__list__eqI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),Ys) )
     => ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_1690_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_1691_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N)
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
tff(fact_1692_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_1693_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_1694_less__numeral__extra_I4_J,axiom,
    ~ aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),one_one_Code_integer) ).

% less_numeral_extra(4)
tff(fact_1695_less__numeral__extra_I4_J,axiom,
    ~ aa_rat_o(ord_less_rat(one_one_rat),one_one_rat) ).

% less_numeral_extra(4)
tff(fact_1696_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_1697_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_1698_neq__NilE,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : L != cons_P2971678138204555879n_assn(X2,Xs2) ) ).

% neq_NilE
tff(fact_1699_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X2: int,Xs2: list_int] : L != cons_int(X2,Xs2) ) ).

% neq_NilE
tff(fact_1700_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] : L != cons_nat(X2,Xs2) ) ).

% neq_NilE
tff(fact_1701_list__2pre__induct,axiom,
    ! [Pa: fun_li7722483182022397486assn_o,W1: list_P8527749157015355191n_assn,W2: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,nil_Pr5671120429643327159n_assn),nil_Pr5671120429643327159n_assn)
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_P8527749157015355191n_assn] :
            ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,W12),W22)
           => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(E,W12)),W22) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_P8527749157015355191n_assn,W23: list_P8527749157015355191n_assn] :
              ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,W13),W23)
             => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,W13),cons_P2971678138204555879n_assn(E,W23)) )
         => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1702_list__2pre__induct,axiom,
    ! [Pa: fun_li3476655817902384897_int_o,W1: list_P8527749157015355191n_assn,W2: list_int] :
      ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,nil_Pr5671120429643327159n_assn),nil_int)
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_int] :
            ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,W12),W22)
           => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(E,W12)),W22) )
       => ( ! [E: int,W13: list_P8527749157015355191n_assn,W23: list_int] :
              ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,W13),W23)
             => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1703_list__2pre__induct,axiom,
    ! [Pa: fun_li3404705325679423581_nat_o,W1: list_P8527749157015355191n_assn,W2: list_nat] :
      ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,nil_Pr5671120429643327159n_assn),nil_nat)
     => ( ! [E: produc6575502325842934193n_assn,W12: list_P8527749157015355191n_assn,W22: list_nat] :
            ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,W12),W22)
           => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(E,W12)),W22) )
       => ( ! [E: nat,W13: list_P8527749157015355191n_assn,W23: list_nat] :
              ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,W13),W23)
             => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1704_list__2pre__induct,axiom,
    ! [Pa: fun_li6324434278616229789assn_o,W1: list_int,W2: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,nil_int),nil_Pr5671120429643327159n_assn)
     => ( ! [E: int,W12: list_int,W22: list_P8527749157015355191n_assn] :
            ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,W12),W22)
           => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(E,W12)),W22) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_int,W23: list_P8527749157015355191n_assn] :
              ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,W13),W23)
             => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,W13),cons_P2971678138204555879n_assn(E,W23)) )
         => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1705_list__2pre__induct,axiom,
    ! [Pa: fun_li1334793589135804498_int_o,W1: list_int,W2: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,nil_int),nil_int)
     => ( ! [E: int,W12: list_int,W22: list_int] :
            ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,W12),W22)
           => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(E,W12)),W22) )
       => ( ! [E: int,W13: list_int,W23: list_int] :
              ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,W13),W23)
             => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1706_list__2pre__induct,axiom,
    ! [Pa: fun_li1262843096912843182_nat_o,W1: list_int,W2: list_nat] :
      ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,nil_int),nil_nat)
     => ( ! [E: int,W12: list_int,W22: list_nat] :
            ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,W12),W22)
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(E,W12)),W22) )
       => ( ! [E: nat,W13: list_int,W23: list_nat] :
              ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,W13),W23)
             => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1707_list__2pre__induct,axiom,
    ! [Pa: fun_li1697075967554290497assn_o,W1: list_nat,W2: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,nil_nat),nil_Pr5671120429643327159n_assn)
     => ( ! [E: nat,W12: list_nat,W22: list_P8527749157015355191n_assn] :
            ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,W12),W22)
           => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(E,W12)),W22) )
       => ( ! [E: produc6575502325842934193n_assn,W13: list_nat,W23: list_P8527749157015355191n_assn] :
              ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,W13),W23)
             => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,W13),cons_P2971678138204555879n_assn(E,W23)) )
         => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1708_list__2pre__induct,axiom,
    ! [Pa: fun_li1097216447485229486_int_o,W1: list_nat,W2: list_int] :
      ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,nil_nat),nil_int)
     => ( ! [E: nat,W12: list_nat,W22: list_int] :
            ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,W12),W22)
           => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(E,W12)),W22) )
       => ( ! [E: int,W13: list_nat,W23: list_int] :
              ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,W13),W23)
             => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,W13),cons_int(E,W23)) )
         => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1709_list__2pre__induct,axiom,
    ! [Pa: fun_li1025265955262268170_nat_o,W1: list_nat,W2: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,nil_nat),nil_nat)
     => ( ! [E: nat,W12: list_nat,W22: list_nat] :
            ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,W12),W22)
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(E,W12)),W22) )
       => ( ! [E: nat,W13: list_nat,W23: list_nat] :
              ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,W13),W23)
             => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,W13),cons_nat(E,W23)) )
         => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,W1),W2) ) ) ) ).

% list_2pre_induct
tff(fact_1710_list__induct__first2,axiom,
    ! [Pa: fun_li2812663585457640420assn_o,Xs: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(Pa,nil_Pr5671120429643327159n_assn)
     => ( ! [X2: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(Pa,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn))
       => ( ! [X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( aa_lis1404944545740551261assn_o(Pa,Xs2)
             => aa_lis1404944545740551261assn_o(Pa,cons_P2971678138204555879n_assn(X1,cons_P2971678138204555879n_assn(X23,Xs2))) )
         => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_1711_list__induct__first2,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( aa_list_int_o(Pa,nil_int)
     => ( ! [X2: int] : aa_list_int_o(Pa,cons_int(X2,nil_int))
       => ( ! [X1: int,X23: int,Xs2: list_int] :
              ( aa_list_int_o(Pa,Xs2)
             => aa_list_int_o(Pa,cons_int(X1,cons_int(X23,Xs2))) )
         => aa_list_int_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_1712_list__induct__first2,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( aa_list_nat_o(Pa,nil_nat)
     => ( ! [X2: nat] : aa_list_nat_o(Pa,cons_nat(X2,nil_nat))
       => ( ! [X1: nat,X23: nat,Xs2: list_nat] :
              ( aa_list_nat_o(Pa,Xs2)
             => aa_list_nat_o(Pa,cons_nat(X1,cons_nat(X23,Xs2))) )
         => aa_list_nat_o(Pa,Xs) ) ) ) ).

% list_induct_first2
tff(fact_1713_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li7722483182022397486assn_o,R: fun_Pr5295903952624206638assn_o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,Xs2),nil_Pr5671120429643327159n_assn)
     => ( ! [Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,nil_Pr5671120429643327159n_assn),Ys3)
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(R,X2),Y2)
             => ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,Xs2),cons_P2971678138204555879n_assn(Y2,Ys3))
               => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ aa_Pro2491721552408928739assn_o(aa_Pro332967637727714143assn_o(R,X2),Y2)
               => ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),Ys3)
                 => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
           => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1714_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li6324434278616229789assn_o,R: fun_in7130107137290581651assn_o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_int] : aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,Xs2),nil_Pr5671120429643327159n_assn)
     => ( ! [Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,nil_int),Ys3)
       => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( aa_Pro2491721552408928739assn_o(aa_int6564194663981188820assn_o(R,X2),Y2)
             => ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,Xs2),cons_P2971678138204555879n_assn(Y2,Ys3))
               => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ aa_Pro2491721552408928739assn_o(aa_int6564194663981188820assn_o(R,X2),Y2)
               => ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(X2,Xs2)),Ys3)
                 => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
           => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1715_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li1697075967554290497assn_o,R: fun_na1082083995841529399assn_o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ! [Xs2: list_nat] : aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,Xs2),nil_Pr5671120429643327159n_assn)
     => ( ! [Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,nil_nat),Ys3)
       => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( aa_Pro2491721552408928739assn_o(aa_nat516171522532136568assn_o(R,X2),Y2)
             => ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,Xs2),cons_P2971678138204555879n_assn(Y2,Ys3))
               => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( ~ aa_Pro2491721552408928739assn_o(aa_nat516171522532136568assn_o(R,X2),Y2)
               => ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(X2,Xs2)),Ys3)
                 => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) ) )
           => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1716_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li3476655817902384897_int_o,R: fun_Pr7575844989570591639_int_o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,Xs2),nil_int)
     => ( ! [Ys3: list_int] : aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,nil_Pr5671120429643327159n_assn),Ys3)
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys3: list_int] :
              ( aa_int_o(aa_Pro5419903954845752976_int_o(R,X2),Y2)
             => ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,Xs2),cons_int(Y2,Ys3))
               => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys3)) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys3: list_int] :
                ( ~ aa_int_o(aa_Pro5419903954845752976_int_o(R,X2),Y2)
               => ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),Ys3)
                 => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys3)) ) )
           => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1717_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li1334793589135804498_int_o,R: fun_int_fun_int_o,Xs: list_int,Ys: list_int] :
      ( ! [Xs2: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs2),nil_int)
     => ( ! [Ys3: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,nil_int),Ys3)
       => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] :
              ( aa_int_o(aa_int_fun_int_o(R,X2),Y2)
             => ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs2),cons_int(Y2,Ys3))
               => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] :
                ( ~ aa_int_o(aa_int_fun_int_o(R,X2),Y2)
               => ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(X2,Xs2)),Ys3)
                 => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)) ) )
           => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1718_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li1097216447485229486_int_o,R: fun_nat_fun_int_o,Xs: list_nat,Ys: list_int] :
      ( ! [Xs2: list_nat] : aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,Xs2),nil_int)
     => ( ! [Ys3: list_int] : aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,nil_nat),Ys3)
       => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int] :
              ( aa_int_o(aa_nat_fun_int_o(R,X2),Y2)
             => ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,Xs2),cons_int(Y2,Ys3))
               => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int] :
                ( ~ aa_int_o(aa_nat_fun_int_o(R,X2),Y2)
               => ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(X2,Xs2)),Ys3)
                 => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)) ) )
           => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1719_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li3404705325679423581_nat_o,R: fun_Pr597617920386261747_nat_o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ! [Xs2: list_P8527749157015355191n_assn] : aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,Xs2),nil_nat)
     => ( ! [Ys3: list_nat] : aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,nil_Pr5671120429643327159n_assn),Ys3)
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys3: list_nat] :
              ( aa_nat_o(aa_Pro7240946300278440940_nat_o(R,X2),Y2)
             => ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,Xs2),cons_nat(Y2,Ys3))
               => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys3: list_nat] :
                ( ~ aa_nat_o(aa_Pro7240946300278440940_nat_o(R,X2),Y2)
               => ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),Ys3)
                 => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
           => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1720_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li1262843096912843182_nat_o,R: fun_int_fun_nat_o,Xs: list_int,Ys: list_nat] :
      ( ! [Xs2: list_int] : aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,Xs2),nil_nat)
     => ( ! [Ys3: list_nat] : aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,nil_int),Ys3)
       => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat] :
              ( aa_nat_o(aa_int_fun_nat_o(R,X2),Y2)
             => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,Xs2),cons_nat(Y2,Ys3))
               => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat] :
                ( ~ aa_nat_o(aa_int_fun_nat_o(R,X2),Y2)
               => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(X2,Xs2)),Ys3)
                 => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1721_mergesort__by__rel__merge__induct,axiom,
    ! [Pa: fun_li1025265955262268170_nat_o,R: fun_nat_fun_nat_o,Xs: list_nat,Ys: list_nat] :
      ( ! [Xs2: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs2),nil_nat)
     => ( ! [Ys3: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,nil_nat),Ys3)
       => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
             => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs2),cons_nat(Y2,Ys3))
               => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
                ( ~ aa_nat_o(aa_nat_fun_nat_o(R,X2),Y2)
               => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(X2,Xs2)),Ys3)
                 => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(X2,Xs2)),cons_nat(Y2,Ys3)) ) )
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_1722_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : nil_Pr5671120429643327159n_assn != cons_P2971678138204555879n_assn(X21,X22) ).

% list.distinct(1)
tff(fact_1723_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] : nil_int != cons_int(X21,X22) ).

% list.distinct(1)
tff(fact_1724_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : nil_nat != cons_nat(X21,X22) ).

% list.distinct(1)
tff(fact_1725_list_OdiscI,axiom,
    ! [List: list_P8527749157015355191n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( List = cons_P2971678138204555879n_assn(X21,X22) )
     => ( List != nil_Pr5671120429643327159n_assn ) ) ).

% list.discI
tff(fact_1726_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List = cons_int(X21,X22) )
     => ( List != nil_int ) ) ).

% list.discI
tff(fact_1727_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List = cons_nat(X21,X22) )
     => ( List != nil_nat ) ) ).

% list.discI
tff(fact_1728_list_Oexhaust,axiom,
    ! [Y: list_P8527749157015355191n_assn] :
      ( ( Y != nil_Pr5671120429643327159n_assn )
     => ~ ! [X212: produc6575502325842934193n_assn,X222: list_P8527749157015355191n_assn] : Y != cons_P2971678138204555879n_assn(X212,X222) ) ).

% list.exhaust
tff(fact_1729_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X222: list_int] : Y != cons_int(X212,X222) ) ).

% list.exhaust
tff(fact_1730_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] : Y != cons_nat(X212,X222) ) ).

% list.exhaust
tff(fact_1731_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X2: int,Xs2: list_int] : X != cons_int(X2,Xs2)
     => ( X = nil_int ) ) ).

% min_list.cases
tff(fact_1732_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs2: list_nat] : X != cons_nat(X2,Xs2)
     => ( X = nil_nat ) ) ).

% min_list.cases
tff(fact_1733_remdups__adj_Ocases,axiom,
    ! [X: list_P8527749157015355191n_assn] :
      ( ( X != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : X != cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn)
       => ~ ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != cons_P2971678138204555879n_assn(X2,cons_P2971678138204555879n_assn(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_1734_remdups__adj_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ( ! [X2: int] : X != cons_int(X2,nil_int)
       => ~ ! [X2: int,Y2: int,Xs2: list_int] : X != cons_int(X2,cons_int(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_1735_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X2: nat] : X != cons_nat(X2,nil_nat)
       => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] : X != cons_nat(X2,cons_nat(Y2,Xs2)) ) ) ).

% remdups_adj.cases
tff(fact_1736_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
    <=> ? [Y3: produc6575502325842934193n_assn,Ys4: list_P8527749157015355191n_assn] : Xs = cons_P2971678138204555879n_assn(Y3,Ys4) ) ).

% neq_Nil_conv
tff(fact_1737_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
    <=> ? [Y3: int,Ys4: list_int] : Xs = cons_int(Y3,Ys4) ) ).

% neq_Nil_conv
tff(fact_1738_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
    <=> ? [Y3: nat,Ys4: list_nat] : Xs = cons_nat(Y3,Ys4) ) ).

% neq_Nil_conv
tff(fact_1739_list__induct2_H,axiom,
    ! [Pa: fun_li7722483182022397486assn_o,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,nil_Pr5671120429643327159n_assn),nil_Pr5671120429643327159n_assn)
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),nil_Pr5671120429643327159n_assn)
       => ( ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,nil_Pr5671120429643327159n_assn),cons_P2971678138204555879n_assn(Y2,Ys3))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,Xs2),Ys3)
               => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) )
           => aa_lis1404944545740551261assn_o(aa_lis8888397383097485663assn_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1740_list__induct2_H,axiom,
    ! [Pa: fun_li3476655817902384897_int_o,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,nil_Pr5671120429643327159n_assn),nil_int)
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),nil_int)
       => ( ! [Y2: int,Ys3: list_int] : aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,nil_Pr5671120429643327159n_assn),cons_int(Y2,Ys3))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: int,Ys3: list_int] :
                ( aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,Xs2),Ys3)
               => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_int(Y2,Ys3)) )
           => aa_list_int_o(aa_lis4145795176574077946_int_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1741_list__induct2_H,axiom,
    ! [Pa: fun_li3404705325679423581_nat_o,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,nil_Pr5671120429643327159n_assn),nil_nat)
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),nil_nat)
       => ( ! [Y2: nat,Ys3: list_nat] : aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,nil_Pr5671120429643327159n_assn),cons_nat(Y2,Ys3))
         => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: nat,Ys3: list_nat] :
                ( aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,Xs2),Ys3)
               => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)),cons_nat(Y2,Ys3)) )
           => aa_list_nat_o(aa_lis6390940144244523862_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1742_list__induct2_H,axiom,
    ! [Pa: fun_li6324434278616229789assn_o,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,nil_int),nil_Pr5671120429643327159n_assn)
     => ( ! [X2: int,Xs2: list_int] : aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(X2,Xs2)),nil_Pr5671120429643327159n_assn)
       => ( ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,nil_int),cons_P2971678138204555879n_assn(Y2,Ys3))
         => ( ! [X2: int,Xs2: list_int,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,Xs2),Ys3)
               => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,cons_int(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) )
           => aa_lis1404944545740551261assn_o(aa_lis2194652677338155742assn_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1743_list__induct2_H,axiom,
    ! [Pa: fun_li1334793589135804498_int_o,Xs: list_int,Ys: list_int] :
      ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,nil_int),nil_int)
     => ( ! [X2: int,Xs2: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(X2,Xs2)),nil_int)
       => ( ! [Y2: int,Ys3: list_int] : aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,nil_int),cons_int(Y2,Ys3))
         => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] :
                ( aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs2),Ys3)
               => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)) )
           => aa_list_int_o(aa_lis7773489400162621755_int_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1744_list__induct2_H,axiom,
    ! [Pa: fun_li1262843096912843182_nat_o,Xs: list_int,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,nil_int),nil_nat)
     => ( ! [X2: int,Xs2: list_int] : aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(X2,Xs2)),nil_nat)
       => ( ! [Y2: nat,Ys3: list_nat] : aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,nil_int),cons_nat(Y2,Ys3))
         => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat] :
                ( aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,Xs2),Ys3)
               => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)) )
           => aa_list_nat_o(aa_lis795262330978291863_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1745_list__induct2_H,axiom,
    ! [Pa: fun_li1697075967554290497assn_o,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,nil_nat),nil_Pr5671120429643327159n_assn)
     => ( ! [X2: nat,Xs2: list_nat] : aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(X2,Xs2)),nil_Pr5671120429643327159n_assn)
       => ( ! [Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,nil_nat),cons_P2971678138204555879n_assn(Y2,Ys3))
         => ( ! [X2: nat,Xs2: list_nat,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
                ( aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,Xs2),Ys3)
               => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,cons_nat(X2,Xs2)),cons_P2971678138204555879n_assn(Y2,Ys3)) )
           => aa_lis1404944545740551261assn_o(aa_lis6790666403130992258assn_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1746_list__induct2_H,axiom,
    ! [Pa: fun_li1097216447485229486_int_o,Xs: list_nat,Ys: list_int] :
      ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,nil_nat),nil_int)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(X2,Xs2)),nil_int)
       => ( ! [Y2: int,Ys3: list_int] : aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,nil_nat),cons_int(Y2,Ys3))
         => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int] :
                ( aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,Xs2),Ys3)
               => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)) )
           => aa_list_int_o(aa_lis7535912258512046743_int_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1747_list__induct2_H,axiom,
    ! [Pa: fun_li1025265955262268170_nat_o,Xs: list_nat,Ys: list_nat] :
      ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,nil_nat),nil_nat)
     => ( ! [X2: nat,Xs2: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(X2,Xs2)),nil_nat)
       => ( ! [Y2: nat,Ys3: list_nat] : aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,nil_nat),cons_nat(Y2,Ys3))
         => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] :
                ( aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs2),Ys3)
               => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,cons_nat(X2,Xs2)),cons_nat(Y2,Ys3)) )
           => aa_list_nat_o(aa_lis557685189327716851_nat_o(Pa,Xs),Ys) ) ) ) ) ).

% list_induct2'
tff(fact_1748_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Pa: fun_li2812663585457640420assn_o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : aa_lis1404944545740551261assn_o(Pa,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn))
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( aa_lis1404944545740551261assn_o(Pa,Xs2)
               => aa_lis1404944545740551261assn_o(Pa,cons_P2971678138204555879n_assn(X2,Xs2)) ) )
         => aa_lis1404944545740551261assn_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_1749_list__nonempty__induct,axiom,
    ! [Xs: list_int,Pa: fun_list_int_o] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : aa_list_int_o(Pa,cons_int(X2,nil_int))
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( aa_list_int_o(Pa,Xs2)
               => aa_list_int_o(Pa,cons_int(X2,Xs2)) ) )
         => aa_list_int_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_1750_list__nonempty__induct,axiom,
    ! [Xs: list_nat,Pa: fun_list_nat_o] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : aa_list_nat_o(Pa,cons_nat(X2,nil_nat))
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( aa_list_nat_o(Pa,Xs2)
               => aa_list_nat_o(Pa,cons_nat(X2,Xs2)) ) )
         => aa_list_nat_o(Pa,Xs) ) ) ) ).

% list_nonempty_induct
tff(fact_1751_merge_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [L2: list_int] : X != produc364263696895485585st_int(nil_int,L2)
     => ( ! [V: int,Va: list_int] : X != produc364263696895485585st_int(cons_int(V,Va),nil_int)
       => ~ ! [X1: int,L1: list_int,X23: int,L2: list_int] : X != produc364263696895485585st_int(cons_int(X1,L1),cons_int(X23,L2)) ) ) ).

% merge.cases
tff(fact_1752_merge_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [L2: list_nat] : X != produc2694037385005941721st_nat(nil_nat,L2)
     => ( ! [V: nat,Va: list_nat] : X != produc2694037385005941721st_nat(cons_nat(V,Va),nil_nat)
       => ~ ! [X1: nat,L1: list_nat,X23: nat,L2: list_nat] : X != produc2694037385005941721st_nat(cons_nat(X1,L1),cons_nat(X23,L2)) ) ) ).

% merge.cases
tff(fact_1753_shuffles_Ocases,axiom,
    ! [X: produc337509097454979815n_assn] :
      ( ! [Ys3: list_P8527749157015355191n_assn] : X != produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,Ys3)
     => ( ! [Xs2: list_P8527749157015355191n_assn] : X != produc3014731801419794263n_assn(Xs2,nil_Pr5671120429643327159n_assn)
       => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : X != produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X2,Xs2),cons_P2971678138204555879n_assn(Y2,Ys3)) ) ) ).

% shuffles.cases
tff(fact_1754_shuffles_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] : X != produc364263696895485585st_int(nil_int,Ys3)
     => ( ! [Xs2: list_int] : X != produc364263696895485585st_int(Xs2,nil_int)
       => ~ ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int] : X != produc364263696895485585st_int(cons_int(X2,Xs2),cons_int(Y2,Ys3)) ) ) ).

% shuffles.cases
tff(fact_1755_shuffles_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] : X != produc2694037385005941721st_nat(nil_nat,Ys3)
     => ( ! [Xs2: list_nat] : X != produc2694037385005941721st_nat(Xs2,nil_nat)
       => ~ ! [X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] : X != produc2694037385005941721st_nat(cons_nat(X2,Xs2),cons_nat(Y2,Ys3)) ) ) ).

% shuffles.cases
tff(fact_1756_successively_Ocases,axiom,
    ! [X: produc3664700245209686448n_assn] :
      ( ! [P3: fun_Pr5295903952624206638assn_o] : X != produc397425096546726432n_assn(P3,nil_Pr5671120429643327159n_assn)
     => ( ! [P3: fun_Pr5295903952624206638assn_o,X2: produc6575502325842934193n_assn] : X != produc397425096546726432n_assn(P3,cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn))
       => ~ ! [P3: fun_Pr5295903952624206638assn_o,X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != produc397425096546726432n_assn(P3,cons_P2971678138204555879n_assn(X2,cons_P2971678138204555879n_assn(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_1757_successively_Ocases,axiom,
    ! [X: produc5262460917026302941st_int] :
      ( ! [P3: fun_int_fun_int_o] : X != produc2087893702315799767st_int(P3,nil_int)
     => ( ! [P3: fun_int_fun_int_o,X2: int] : X != produc2087893702315799767st_int(P3,cons_int(X2,nil_int))
       => ~ ! [P3: fun_int_fun_int_o,X2: int,Y2: int,Xs2: list_int] : X != produc2087893702315799767st_int(P3,cons_int(X2,cons_int(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_1758_successively_Ocases,axiom,
    ! [X: produc7317866342734830137st_nat] :
      ( ! [P3: fun_nat_fun_nat_o] : X != produc5831067002601796915st_nat(P3,nil_nat)
     => ( ! [P3: fun_nat_fun_nat_o,X2: nat] : X != produc5831067002601796915st_nat(P3,cons_nat(X2,nil_nat))
       => ~ ! [P3: fun_nat_fun_nat_o,X2: nat,Y2: nat,Xs2: list_nat] : X != produc5831067002601796915st_nat(P3,cons_nat(X2,cons_nat(Y2,Xs2))) ) ) ).

% successively.cases
tff(fact_1759_sorted__wrt_Ocases,axiom,
    ! [X: produc3664700245209686448n_assn] :
      ( ! [P3: fun_Pr5295903952624206638assn_o] : X != produc397425096546726432n_assn(P3,nil_Pr5671120429643327159n_assn)
     => ~ ! [P3: fun_Pr5295903952624206638assn_o,X2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : X != produc397425096546726432n_assn(P3,cons_P2971678138204555879n_assn(X2,Ys3)) ) ).

% sorted_wrt.cases
tff(fact_1760_sorted__wrt_Ocases,axiom,
    ! [X: produc5262460917026302941st_int] :
      ( ! [P3: fun_int_fun_int_o] : X != produc2087893702315799767st_int(P3,nil_int)
     => ~ ! [P3: fun_int_fun_int_o,X2: int,Ys3: list_int] : X != produc2087893702315799767st_int(P3,cons_int(X2,Ys3)) ) ).

% sorted_wrt.cases
tff(fact_1761_sorted__wrt_Ocases,axiom,
    ! [X: produc7317866342734830137st_nat] :
      ( ! [P3: fun_nat_fun_nat_o] : X != produc5831067002601796915st_nat(P3,nil_nat)
     => ~ ! [P3: fun_nat_fun_nat_o,X2: nat,Ys3: list_nat] : X != produc5831067002601796915st_nat(P3,cons_nat(X2,Ys3)) ) ).

% sorted_wrt.cases
tff(fact_1762_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc858023921370090807n_assn] :
      ( ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn] : X != produc7497315745449715623n_assn(produc3014731801419794263n_assn(Xs1,Xs22),nil_Pr5671120429643327159n_assn)
     => ( ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn] : X != produc7497315745449715623n_assn(produc3014731801419794263n_assn(Xs1,Xs22),cons_P2971678138204555879n_assn(X2,nil_Pr5671120429643327159n_assn))
       => ~ ! [Xs1: list_P8527749157015355191n_assn,Xs22: list_P8527749157015355191n_assn,X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != produc7497315745449715623n_assn(produc3014731801419794263n_assn(Xs1,Xs22),cons_P2971678138204555879n_assn(X1,cons_P2971678138204555879n_assn(X23,Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_1763_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc5625914783551145028st_int] :
      ( ! [Xs1: list_int,Xs22: list_int] : X != produc6571967182415643454st_int(produc364263696895485585st_int(Xs1,Xs22),nil_int)
     => ( ! [Xs1: list_int,Xs22: list_int,X2: int] : X != produc6571967182415643454st_int(produc364263696895485585st_int(Xs1,Xs22),cons_int(X2,nil_int))
       => ~ ! [Xs1: list_int,Xs22: list_int,X1: int,X23: int,Xs2: list_int] : X != produc6571967182415643454st_int(produc364263696895485585st_int(Xs1,Xs22),cons_int(X1,cons_int(X23,Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_1764_mergesort__by__rel__split_Ocases,axiom,
    ! [X: produc8368264342088597808st_nat] :
      ( ! [Xs1: list_nat,Xs22: list_nat] : X != produc1778712578675790378st_nat(produc2694037385005941721st_nat(Xs1,Xs22),nil_nat)
     => ( ! [Xs1: list_nat,Xs22: list_nat,X2: nat] : X != produc1778712578675790378st_nat(produc2694037385005941721st_nat(Xs1,Xs22),cons_nat(X2,nil_nat))
       => ~ ! [Xs1: list_nat,Xs22: list_nat,X1: nat,X23: nat,Xs2: list_nat] : X != produc1778712578675790378st_nat(produc2694037385005941721st_nat(Xs1,Xs22),cons_nat(X1,cons_nat(X23,Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_1765_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc3761461193743331680n_assn] :
      ( ! [R3: fun_Pr5295903952624206638assn_o,X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(R3,produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X2,Xs2),cons_P2971678138204555879n_assn(Y2,Ys3)))
     => ( ! [R3: fun_Pr5295903952624206638assn_o,Xs2: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(R3,produc3014731801419794263n_assn(Xs2,nil_Pr5671120429643327159n_assn))
       => ~ ! [R3: fun_Pr5295903952624206638assn_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(R3,produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,cons_P2971678138204555879n_assn(V,Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_1766_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc8539026191236913166st_int] :
      ( ! [R3: fun_int_fun_int_o,X2: int,Xs2: list_int,Y2: int,Ys3: list_int] : X != produc1437073085925314174st_int(R3,produc364263696895485585st_int(cons_int(X2,Xs2),cons_int(Y2,Ys3)))
     => ( ! [R3: fun_int_fun_int_o,Xs2: list_int] : X != produc1437073085925314174st_int(R3,produc364263696895485585st_int(Xs2,nil_int))
       => ~ ! [R3: fun_int_fun_int_o,V: int,Va: list_int] : X != produc1437073085925314174st_int(R3,produc364263696895485585st_int(nil_int,cons_int(V,Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_1767_mergesort__by__rel__merge_Ocases,axiom,
    ! [X: produc3739422418112584862st_nat] :
      ( ! [R3: fun_nat_fun_nat_o,X2: nat,Xs2: list_nat,Y2: nat,Ys3: list_nat] : X != produc3816665474402891534st_nat(R3,produc2694037385005941721st_nat(cons_nat(X2,Xs2),cons_nat(Y2,Ys3)))
     => ( ! [R3: fun_nat_fun_nat_o,Xs2: list_nat] : X != produc3816665474402891534st_nat(R3,produc2694037385005941721st_nat(Xs2,nil_nat))
       => ~ ! [R3: fun_nat_fun_nat_o,V: nat,Va: list_nat] : X != produc3816665474402891534st_nat(R3,produc2694037385005941721st_nat(nil_nat,cons_nat(V,Va))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_1768_quicksort__by__rel_Ocases,axiom,
    ! [X: produc3761461193743331680n_assn] :
      ( ! [R3: fun_Pr5295903952624206638assn_o,Sl: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(R3,produc3014731801419794263n_assn(Sl,nil_Pr5671120429643327159n_assn))
     => ~ ! [R3: fun_Pr5295903952624206638assn_o,Sl: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(R3,produc3014731801419794263n_assn(Sl,cons_P2971678138204555879n_assn(X2,Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_1769_quicksort__by__rel_Ocases,axiom,
    ! [X: produc8539026191236913166st_int] :
      ( ! [R3: fun_int_fun_int_o,Sl: list_int] : X != produc1437073085925314174st_int(R3,produc364263696895485585st_int(Sl,nil_int))
     => ~ ! [R3: fun_int_fun_int_o,Sl: list_int,X2: int,Xs2: list_int] : X != produc1437073085925314174st_int(R3,produc364263696895485585st_int(Sl,cons_int(X2,Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_1770_quicksort__by__rel_Ocases,axiom,
    ! [X: produc3739422418112584862st_nat] :
      ( ! [R3: fun_nat_fun_nat_o,Sl: list_nat] : X != produc3816665474402891534st_nat(R3,produc2694037385005941721st_nat(Sl,nil_nat))
     => ~ ! [R3: fun_nat_fun_nat_o,Sl: list_nat,X2: nat,Xs2: list_nat] : X != produc3816665474402891534st_nat(R3,produc2694037385005941721st_nat(Sl,cons_nat(X2,Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_1771_partition__rev_Ocases,axiom,
    ! [X: produc8404423329169178818n_assn] :
      ( ! [P3: fun_Pr3571187171380813162assn_o,Yes: list_P8527749157015355191n_assn,No: list_P8527749157015355191n_assn] : X != produc21272824405065460n_assn(P3,produc7497315745449715623n_assn(produc3014731801419794263n_assn(Yes,No),nil_Pr5671120429643327159n_assn))
     => ~ ! [P3: fun_Pr3571187171380813162assn_o,Yes: list_P8527749157015355191n_assn,No: list_P8527749157015355191n_assn,X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] : X != produc21272824405065460n_assn(P3,produc7497315745449715623n_assn(produc3014731801419794263n_assn(Yes,No),cons_P2971678138204555879n_assn(X2,Xs2))) ) ).

% partition_rev.cases
tff(fact_1772_partition__rev_Ocases,axiom,
    ! [X: produc8015205931655835320st_int] :
      ( ! [P3: fun_int_o,Yes: list_int,No: list_int] : X != produc5853528864671902002st_int(P3,produc6571967182415643454st_int(produc364263696895485585st_int(Yes,No),nil_int))
     => ~ ! [P3: fun_int_o,Yes: list_int,No: list_int,X2: int,Xs2: list_int] : X != produc5853528864671902002st_int(P3,produc6571967182415643454st_int(produc364263696895485585st_int(Yes,No),cons_int(X2,Xs2))) ) ).

% partition_rev.cases
tff(fact_1773_partition__rev_Ocases,axiom,
    ! [X: produc4805982502724343552st_nat] :
      ( ! [P3: fun_nat_o,Yes: list_nat,No: list_nat] : X != produc5141916181857879930st_nat(P3,produc1778712578675790378st_nat(produc2694037385005941721st_nat(Yes,No),nil_nat))
     => ~ ! [P3: fun_nat_o,Yes: list_nat,No: list_nat,X2: nat,Xs2: list_nat] : X != produc5141916181857879930st_nat(P3,produc1778712578675790378st_nat(produc2694037385005941721st_nat(Yes,No),cons_nat(X2,Xs2))) ) ).

% partition_rev.cases
tff(fact_1774_list__all__zip_Ocases,axiom,
    ! [X: produc3761461193743331680n_assn] :
      ( ! [P3: fun_Pr5295903952624206638assn_o] : X != produc4364101221923024848n_assn(P3,produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,nil_Pr5671120429643327159n_assn))
     => ( ! [P3: fun_Pr5295903952624206638assn_o,A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B3: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(P3,produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(A4,As),cons_P2971678138204555879n_assn(B3,Bs2)))
       => ( ! [P3: fun_Pr5295903952624206638assn_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(P3,produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(V,Va),nil_Pr5671120429643327159n_assn))
         => ~ ! [P3: fun_Pr5295903952624206638assn_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc4364101221923024848n_assn(P3,produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,cons_P2971678138204555879n_assn(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1775_list__all__zip_Ocases,axiom,
    ! [X: produc1706455236302107290st_int] :
      ( ! [P3: fun_Pr7575844989570591639_int_o] : X != produc3210926427610269138st_int(P3,produc1996262876970182428st_int(nil_Pr5671120429643327159n_assn,nil_int))
     => ( ! [P3: fun_Pr7575844989570591639_int_o,A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B3: int,Bs2: list_int] : X != produc3210926427610269138st_int(P3,produc1996262876970182428st_int(cons_P2971678138204555879n_assn(A4,As),cons_int(B3,Bs2)))
       => ( ! [P3: fun_Pr7575844989570591639_int_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc3210926427610269138st_int(P3,produc1996262876970182428st_int(cons_P2971678138204555879n_assn(V,Va),nil_int))
         => ~ ! [P3: fun_Pr7575844989570591639_int_o,V: int,Va: list_int] : X != produc3210926427610269138st_int(P3,produc1996262876970182428st_int(nil_Pr5671120429643327159n_assn,cons_int(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1776_list__all__zip_Ocases,axiom,
    ! [X: produc3066041413738551194st_nat] :
      ( ! [P3: fun_Pr597617920386261747_nat_o] : X != produc2247047331190521042st_nat(P3,produc6174113896479379136st_nat(nil_Pr5671120429643327159n_assn,nil_nat))
     => ( ! [P3: fun_Pr597617920386261747_nat_o,A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn,B3: nat,Bs2: list_nat] : X != produc2247047331190521042st_nat(P3,produc6174113896479379136st_nat(cons_P2971678138204555879n_assn(A4,As),cons_nat(B3,Bs2)))
       => ( ! [P3: fun_Pr597617920386261747_nat_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc2247047331190521042st_nat(P3,produc6174113896479379136st_nat(cons_P2971678138204555879n_assn(V,Va),nil_nat))
         => ~ ! [P3: fun_Pr597617920386261747_nat_o,V: nat,Va: list_nat] : X != produc2247047331190521042st_nat(P3,produc6174113896479379136st_nat(nil_Pr5671120429643327159n_assn,cons_nat(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1777_list__all__zip_Ocases,axiom,
    ! [X: produc4695431100945413732n_assn] :
      ( ! [P3: fun_in7130107137290581651assn_o] : X != produc2124837482592309404n_assn(P3,produc4510294837168919970n_assn(nil_int,nil_Pr5671120429643327159n_assn))
     => ( ! [P3: fun_in7130107137290581651assn_o,A4: int,As: list_int,B3: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] : X != produc2124837482592309404n_assn(P3,produc4510294837168919970n_assn(cons_int(A4,As),cons_P2971678138204555879n_assn(B3,Bs2)))
       => ( ! [P3: fun_in7130107137290581651assn_o,V: int,Va: list_int] : X != produc2124837482592309404n_assn(P3,produc4510294837168919970n_assn(cons_int(V,Va),nil_Pr5671120429643327159n_assn))
         => ~ ! [P3: fun_in7130107137290581651assn_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc2124837482592309404n_assn(P3,produc4510294837168919970n_assn(nil_int,cons_P2971678138204555879n_assn(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1778_list__all__zip_Ocases,axiom,
    ! [X: produc8539026191236913166st_int] :
      ( ! [P3: fun_int_fun_int_o] : X != produc1437073085925314174st_int(P3,produc364263696895485585st_int(nil_int,nil_int))
     => ( ! [P3: fun_int_fun_int_o,A4: int,As: list_int,B3: int,Bs2: list_int] : X != produc1437073085925314174st_int(P3,produc364263696895485585st_int(cons_int(A4,As),cons_int(B3,Bs2)))
       => ( ! [P3: fun_int_fun_int_o,V: int,Va: list_int] : X != produc1437073085925314174st_int(P3,produc364263696895485585st_int(cons_int(V,Va),nil_int))
         => ~ ! [P3: fun_int_fun_int_o,V: int,Va: list_int] : X != produc1437073085925314174st_int(P3,produc364263696895485585st_int(nil_int,cons_int(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1779_list__all__zip_Ocases,axiom,
    ! [X: produc7479177255817211222st_nat] :
      ( ! [P3: fun_int_fun_nat_o] : X != produc7277130913504196038st_nat(P3,produc4542114716404682293st_nat(nil_int,nil_nat))
     => ( ! [P3: fun_int_fun_nat_o,A4: int,As: list_int,B3: nat,Bs2: list_nat] : X != produc7277130913504196038st_nat(P3,produc4542114716404682293st_nat(cons_int(A4,As),cons_nat(B3,Bs2)))
       => ( ! [P3: fun_int_fun_nat_o,V: int,Va: list_int] : X != produc7277130913504196038st_nat(P3,produc4542114716404682293st_nat(cons_int(V,Va),nil_nat))
         => ~ ! [P3: fun_int_fun_nat_o,V: nat,Va: list_nat] : X != produc7277130913504196038st_nat(P3,produc4542114716404682293st_nat(nil_int,cons_nat(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1780_list__all__zip_Ocases,axiom,
    ! [X: produc348787858823015268n_assn] :
      ( ! [P3: fun_na1082083995841529399assn_o] : X != produc7273117628184114076n_assn(P3,produc7706768604797473534n_assn(nil_nat,nil_Pr5671120429643327159n_assn))
     => ( ! [P3: fun_na1082083995841529399assn_o,A4: nat,As: list_nat,B3: produc6575502325842934193n_assn,Bs2: list_P8527749157015355191n_assn] : X != produc7273117628184114076n_assn(P3,produc7706768604797473534n_assn(cons_nat(A4,As),cons_P2971678138204555879n_assn(B3,Bs2)))
       => ( ! [P3: fun_na1082083995841529399assn_o,V: nat,Va: list_nat] : X != produc7273117628184114076n_assn(P3,produc7706768604797473534n_assn(cons_nat(V,Va),nil_Pr5671120429643327159n_assn))
         => ~ ! [P3: fun_na1082083995841529399assn_o,V: produc6575502325842934193n_assn,Va: list_P8527749157015355191n_assn] : X != produc7273117628184114076n_assn(P3,produc7706768604797473534n_assn(nil_nat,cons_P2971678138204555879n_assn(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1781_list__all__zip_Ocases,axiom,
    ! [X: produc4799271353532286806st_int] :
      ( ! [P3: fun_nat_fun_int_o] : X != produc7199979683678785478st_int(P3,produc7739558402351520821st_int(nil_nat,nil_int))
     => ( ! [P3: fun_nat_fun_int_o,A4: nat,As: list_nat,B3: int,Bs2: list_int] : X != produc7199979683678785478st_int(P3,produc7739558402351520821st_int(cons_nat(A4,As),cons_int(B3,Bs2)))
       => ( ! [P3: fun_nat_fun_int_o,V: nat,Va: list_nat] : X != produc7199979683678785478st_int(P3,produc7739558402351520821st_int(cons_nat(V,Va),nil_int))
         => ~ ! [P3: fun_nat_fun_int_o,V: int,Va: list_int] : X != produc7199979683678785478st_int(P3,produc7739558402351520821st_int(nil_nat,cons_int(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1782_list__all__zip_Ocases,axiom,
    ! [X: produc3739422418112584862st_nat] :
      ( ! [P3: fun_nat_fun_nat_o] : X != produc3816665474402891534st_nat(P3,produc2694037385005941721st_nat(nil_nat,nil_nat))
     => ( ! [P3: fun_nat_fun_nat_o,A4: nat,As: list_nat,B3: nat,Bs2: list_nat] : X != produc3816665474402891534st_nat(P3,produc2694037385005941721st_nat(cons_nat(A4,As),cons_nat(B3,Bs2)))
       => ( ! [P3: fun_nat_fun_nat_o,V: nat,Va: list_nat] : X != produc3816665474402891534st_nat(P3,produc2694037385005941721st_nat(cons_nat(V,Va),nil_nat))
         => ~ ! [P3: fun_nat_fun_nat_o,V: nat,Va: list_nat] : X != produc3816665474402891534st_nat(P3,produc2694037385005941721st_nat(nil_nat,cons_nat(V,Va))) ) ) ) ).

% list_all_zip.cases
tff(fact_1783_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,B)),A) ) ).

% minus_less_iff
tff(fact_1784_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(A)),B)
    <=> aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),A) ) ).

% minus_less_iff
tff(fact_1785_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),A) ) ).

% minus_less_iff
tff(fact_1786_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(uminus_uminus_int,B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(uminus_uminus_int,A)) ) ).

% less_minus_iff
tff(fact_1787_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),uminus_uminus_rat(B))
    <=> aa_rat_o(ord_less_rat(B),uminus_uminus_rat(A)) ) ).

% less_minus_iff
tff(fact_1788_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ).

% less_minus_iff
tff(fact_1789_measures__less,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
     => member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(cons_fun_nat_nat(F,Fs))) ) ).

% measures_less
tff(fact_1790_measures__less,axiom,
    ! [F: fun_int_nat,X: int,Y: int,Fs: list_fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_int_nat(F,X)),aa_int_nat(F,Y))
     => member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F,Fs))) ) ).

% measures_less
tff(fact_1791_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1792_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1793_less__infI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI1
tff(fact_1794_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI1
tff(fact_1795_less__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),X)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI1
tff(fact_1796_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI1
tff(fact_1797_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI1
tff(fact_1798_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),X)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1799_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),X)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1800_less__infI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,X: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),X)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),X) ) ).

% less_infI2
tff(fact_1801_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),X)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),X) ) ).

% less_infI2
tff(fact_1802_less__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),X)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),X) ) ).

% less_infI2
tff(fact_1803_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),X)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),X) ) ).

% less_infI2
tff(fact_1804_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),X)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),X) ) ).

% less_infI2
tff(fact_1805_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1806_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1807_inf_Oabsorb3,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1808_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1809_inf_Oabsorb3,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1810_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1811_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = A ) ) ).

% inf.absorb3
tff(fact_1812_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),A)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1813_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1814_inf_Oabsorb4,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),A)
     => ( aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1815_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1816_inf_Oabsorb4,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1817_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1818_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) = B ) ) ).

% inf.absorb4
tff(fact_1819_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B),C))
     => ~ ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
         => ~ aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1820_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B),C))
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1821_inf_Ostrict__boundedE,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,B),C))
     => ~ ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
         => ~ aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1822_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,B),C))
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1823_inf_Ostrict__boundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,B),C))
     => ~ ( aa_rat_o(ord_less_rat(A),B)
         => ~ aa_rat_o(ord_less_rat(A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1824_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,B),C))
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1825_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(inf_inf_int,B),C))
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,A),C) ) ) ).

% inf.strict_boundedE
tff(fact_1826_inf_Ostrict__order__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),B)
    <=> ( ( A = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1827_inf_Ostrict__order__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1828_inf_Ostrict__order__iff,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),B)
    <=> ( ( A = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1829_inf_Ostrict__order__iff,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1830_inf_Ostrict__order__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1831_inf_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1832_inf_Ostrict__order__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B) )
        & ( A != B ) ) ) ).

% inf.strict_order_iff
tff(fact_1833_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,A),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1834_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1835_inf_Ostrict__coboundedI1,axiom,
    ! [A: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,B: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,A),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1836_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),C)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1837_inf_Ostrict__coboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),C)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1838_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1839_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI1
tff(fact_1840_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,B),C)
     => aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1841_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),C)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1842_inf_Ostrict__coboundedI2,axiom,
    ! [B: fun_Pr3238374608701176162_int_o,C: fun_Pr3238374608701176162_int_o,A: fun_Pr3238374608701176162_int_o] :
      ( aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,B),C)
     => aa_fun2484551872032894584nt_o_o(aa_fun6202611221056298223nt_o_o(ord_le7515035663206590902_int_o,aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1843_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),C)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1844_inf_Ostrict__coboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),C)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(inf_inf_rat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1845_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(inf_inf_nat,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1846_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(inf_inf_int,A),B)),C) ) ).

% inf.strict_coboundedI2
tff(fact_1847_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1848_sup_Ostrict__coboundedI2,axiom,
    ! [C: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1849_sup_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),B)
     => aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1850_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1851_sup_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI2
tff(fact_1852_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1853_sup_Ostrict__coboundedI1,axiom,
    ! [C: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1854_sup_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),A)
     => aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1855_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1856_sup_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% sup.strict_coboundedI1
tff(fact_1857_sup_Ostrict__order__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
    <=> ( ( A = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1858_sup_Ostrict__order__iff,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
    <=> ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1859_sup_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1860_sup_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1861_sup_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) )
        & ( A != B ) ) ) ).

% sup.strict_order_iff
tff(fact_1862_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B),C)),A)
     => ~ ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
         => ~ aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_1863_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,B),C)),A)
     => ~ ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
         => ~ aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_1864_sup_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,B),C)),A)
     => ~ ( aa_rat_o(ord_less_rat(B),A)
         => ~ aa_rat_o(ord_less_rat(C),A) ) ) ).

% sup.strict_boundedE
tff(fact_1865_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_1866_sup_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(sup_sup_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% sup.strict_boundedE
tff(fact_1867_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,A),B)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1868_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,A),B)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1869_sup_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1870_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1871_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = B ) ) ).

% sup.absorb4
tff(fact_1872_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,B),A)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1873_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,B),A)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1874_sup_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1875_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1876_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B) = A ) ) ).

% sup.absorb3
tff(fact_1877_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),B)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI2
tff(fact_1878_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),B)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI2
tff(fact_1879_less__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(X),B)
     => aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI2
tff(fact_1880_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI2
tff(fact_1881_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI2
tff(fact_1882_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),A)
     => aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,X),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,A),B)) ) ).

% less_supI1
tff(fact_1883_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),A)
     => aa_assn_o(aa_assn_fun_assn_o(ord_less_assn,X),aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,A),B)) ) ).

% less_supI1
tff(fact_1884_less__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(X),A)
     => aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(sup_sup_rat,A),B)) ) ).

% less_supI1
tff(fact_1885_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(sup_sup_nat,A),B)) ) ).

% less_supI1
tff(fact_1886_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,X),aa_int_int(aa_int_fun_int_int(sup_sup_int,A),B)) ) ).

% less_supI1
tff(fact_1887_map__eq__Cons__conv,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = cons_assn(Y,Ys) )
    <=> ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs = cons_P2971678138204555879n_assn(Z3,Zs2) )
          & ( aa_Pro2979818503705714077n_assn(F,Z3) = Y )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs2) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_1888_map__eq__Cons__conv,axiom,
    ! [F: fun_int_int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( map_int_int(F,Xs) = cons_int(Y,Ys) )
    <=> ? [Z3: int,Zs2: list_int] :
          ( ( Xs = cons_int(Z3,Zs2) )
          & ( aa_int_int(F,Z3) = Y )
          & ( map_int_int(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_1889_map__eq__Cons__conv,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( map_nat_int(F,Xs) = cons_int(Y,Ys) )
    <=> ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs = cons_nat(Z3,Zs2) )
          & ( aa_nat_int(F,Z3) = Y )
          & ( map_nat_int(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_1890_map__eq__Cons__conv,axiom,
    ! [F: fun_int_nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = cons_nat(Y,Ys) )
    <=> ? [Z3: int,Zs2: list_int] :
          ( ( Xs = cons_int(Z3,Zs2) )
          & ( aa_int_nat(F,Z3) = Y )
          & ( map_int_nat(F,Zs2) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_1891_map__eq__Cons__conv,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = cons_nat(Y,Ys) )
    <=> ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs = cons_nat(Z3,Zs2) )
          & ( aa_nat_nat(F,Z3) = Y )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs2) = Ys ) ) ) ).

% map_eq_Cons_conv
tff(fact_1892_Cons__eq__map__conv,axiom,
    ! [X: assn,Xs: list_assn,F: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( cons_assn(X,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ys) )
    <=> ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys = cons_P2971678138204555879n_assn(Z3,Zs2) )
          & ( X = aa_Pro2979818503705714077n_assn(F,Z3) )
          & ( Xs = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs2) ) ) ) ).

% Cons_eq_map_conv
tff(fact_1893_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: fun_int_int,Ys: list_int] :
      ( ( cons_int(X,Xs) = map_int_int(F,Ys) )
    <=> ? [Z3: int,Zs2: list_int] :
          ( ( Ys = cons_int(Z3,Zs2) )
          & ( X = aa_int_int(F,Z3) )
          & ( Xs = map_int_int(F,Zs2) ) ) ) ).

% Cons_eq_map_conv
tff(fact_1894_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: fun_nat_int,Ys: list_nat] :
      ( ( cons_int(X,Xs) = map_nat_int(F,Ys) )
    <=> ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys = cons_nat(Z3,Zs2) )
          & ( X = aa_nat_int(F,Z3) )
          & ( Xs = map_nat_int(F,Zs2) ) ) ) ).

% Cons_eq_map_conv
tff(fact_1895_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: fun_int_nat,Ys: list_int] :
      ( ( cons_nat(X,Xs) = map_int_nat(F,Ys) )
    <=> ? [Z3: int,Zs2: list_int] :
          ( ( Ys = cons_int(Z3,Zs2) )
          & ( X = aa_int_nat(F,Z3) )
          & ( Xs = map_int_nat(F,Zs2) ) ) ) ).

% Cons_eq_map_conv
tff(fact_1896_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_nat,Ys: list_nat] :
      ( ( cons_nat(X,Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ys) )
    <=> ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys = cons_nat(Z3,Zs2) )
          & ( X = aa_nat_nat(F,Z3) )
          & ( Xs = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs2) ) ) ) ).

% Cons_eq_map_conv
tff(fact_1897_map__eq__Cons__D,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = cons_assn(Y,Ys) )
     => ? [Z2: produc6575502325842934193n_assn,Zs3: list_P8527749157015355191n_assn] :
          ( ( Xs = cons_P2971678138204555879n_assn(Z2,Zs3) )
          & ( aa_Pro2979818503705714077n_assn(F,Z2) = Y )
          & ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs3) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_1898_map__eq__Cons__D,axiom,
    ! [F: fun_int_int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( map_int_int(F,Xs) = cons_int(Y,Ys) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Xs = cons_int(Z2,Zs3) )
          & ( aa_int_int(F,Z2) = Y )
          & ( map_int_int(F,Zs3) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_1899_map__eq__Cons__D,axiom,
    ! [F: fun_nat_int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( map_nat_int(F,Xs) = cons_int(Y,Ys) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Xs = cons_nat(Z2,Zs3) )
          & ( aa_nat_int(F,Z2) = Y )
          & ( map_nat_int(F,Zs3) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_1900_map__eq__Cons__D,axiom,
    ! [F: fun_int_nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = cons_nat(Y,Ys) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Xs = cons_int(Z2,Zs3) )
          & ( aa_int_nat(F,Z2) = Y )
          & ( map_int_nat(F,Zs3) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_1901_map__eq__Cons__D,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = cons_nat(Y,Ys) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Xs = cons_nat(Z2,Zs3) )
          & ( aa_nat_nat(F,Z2) = Y )
          & ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs3) = Ys ) ) ) ).

% map_eq_Cons_D
tff(fact_1902_Cons__eq__map__D,axiom,
    ! [X: assn,Xs: list_assn,F: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( cons_assn(X,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Ys) )
     => ? [Z2: produc6575502325842934193n_assn,Zs3: list_P8527749157015355191n_assn] :
          ( ( Ys = cons_P2971678138204555879n_assn(Z2,Zs3) )
          & ( X = aa_Pro2979818503705714077n_assn(F,Z2) )
          & ( Xs = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Zs3) ) ) ) ).

% Cons_eq_map_D
tff(fact_1903_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: fun_int_int,Ys: list_int] :
      ( ( cons_int(X,Xs) = map_int_int(F,Ys) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Ys = cons_int(Z2,Zs3) )
          & ( X = aa_int_int(F,Z2) )
          & ( Xs = map_int_int(F,Zs3) ) ) ) ).

% Cons_eq_map_D
tff(fact_1904_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: fun_nat_int,Ys: list_nat] :
      ( ( cons_int(X,Xs) = map_nat_int(F,Ys) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Ys = cons_nat(Z2,Zs3) )
          & ( X = aa_nat_int(F,Z2) )
          & ( Xs = map_nat_int(F,Zs3) ) ) ) ).

% Cons_eq_map_D
tff(fact_1905_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: fun_int_nat,Ys: list_int] :
      ( ( cons_nat(X,Xs) = map_int_nat(F,Ys) )
     => ? [Z2: int,Zs3: list_int] :
          ( ( Ys = cons_int(Z2,Zs3) )
          & ( X = aa_int_nat(F,Z2) )
          & ( Xs = map_int_nat(F,Zs3) ) ) ) ).

% Cons_eq_map_D
tff(fact_1906_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: fun_nat_nat,Ys: list_nat] :
      ( ( cons_nat(X,Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Ys) )
     => ? [Z2: nat,Zs3: list_nat] :
          ( ( Ys = cons_nat(Z2,Zs3) )
          & ( X = aa_nat_nat(F,Z2) )
          & ( Xs = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Zs3) ) ) ) ).

% Cons_eq_map_D
tff(fact_1907_list_Osimps_I9_J,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] : aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),cons_P2971678138204555879n_assn(X21,X22)) = cons_assn(aa_Pro2979818503705714077n_assn(F,X21),aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),X22)) ).

% list.simps(9)
tff(fact_1908_list_Osimps_I9_J,axiom,
    ! [F: fun_int_int,X21: int,X22: list_int] : map_int_int(F,cons_int(X21,X22)) = cons_int(aa_int_int(F,X21),map_int_int(F,X22)) ).

% list.simps(9)
tff(fact_1909_list_Osimps_I9_J,axiom,
    ! [F: fun_int_nat,X21: int,X22: list_int] : map_int_nat(F,cons_int(X21,X22)) = cons_nat(aa_int_nat(F,X21),map_int_nat(F,X22)) ).

% list.simps(9)
tff(fact_1910_list_Osimps_I9_J,axiom,
    ! [F: fun_nat_int,X21: nat,X22: list_nat] : map_nat_int(F,cons_nat(X21,X22)) = cons_int(aa_nat_int(F,X21),map_nat_int(F,X22)) ).

% list.simps(9)
tff(fact_1911_list_Osimps_I9_J,axiom,
    ! [F: fun_nat_nat,X21: nat,X22: list_nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),cons_nat(X21,X22)) = cons_nat(aa_nat_nat(F,X21),aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),X22)) ).

% list.simps(9)
tff(fact_1912_splice_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] : splice_int(cons_int(X,Xs),Ys) = cons_int(X,splice_int(Ys,Xs)) ).

% splice.simps(2)
tff(fact_1913_splice_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] : splice_nat(cons_nat(X,Xs),Ys) = cons_nat(X,splice_nat(Ys,Xs)) ).

% splice.simps(2)
tff(fact_1914_null__rec_I1_J,axiom,
    ! [X: int,Xs: list_int] : ~ aa_list_int_o(null_int,cons_int(X,Xs)) ).

% null_rec(1)
tff(fact_1915_null__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat] : ~ aa_list_nat_o(null_nat,cons_nat(X,Xs)) ).

% null_rec(1)
tff(fact_1916_member__rec_I1_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( aa_int_o(member_int(cons_int(X,Xs)),Y)
    <=> ( ( X = Y )
        | aa_int_o(member_int(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_1917_member__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( aa_nat_o(member_nat(cons_nat(X,Xs)),Y)
    <=> ( ( X = Y )
        | aa_nat_o(member_nat(Xs),Y) ) ) ).

% member_rec(1)
tff(fact_1918_mult__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1919_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1920_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),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,A),B)) ) ) ).

% mult_neg_neg
tff(fact_1921_not__square__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),zero_z3403309356797280102nteger) ).

% not_square_less_zero
tff(fact_1922_not__square__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),A)),zero_zero_rat) ).

% not_square_less_zero
tff(fact_1923_not__square__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),zero_zero_int) ).

% not_square_less_zero
tff(fact_1924_mult__less__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger)
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger) )
        | ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1925_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat)
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_rat(B),zero_zero_rat) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1926_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int)
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ) ).

% mult_less_0_iff
tff(fact_1927_mult__neg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_neg_pos
tff(fact_1928_mult__neg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_neg_pos
tff(fact_1929_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_neg_pos
tff(fact_1930_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),zero_zero_nat) ) ) ).

% mult_neg_pos
tff(fact_1931_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),zero_zero_int) ) ) ).

% mult_neg_pos
tff(fact_1932_mult__pos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% mult_pos_neg
tff(fact_1933_mult__pos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg
tff(fact_1934_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),zero_zero_rat) ) ) ).

% mult_pos_neg
tff(fact_1935_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% mult_pos_neg
tff(fact_1936_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),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,A),B)),zero_zero_int) ) ) ).

% mult_pos_neg
tff(fact_1937_mult__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1938_mult__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1939_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1940_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1941_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% mult_pos_pos
tff(fact_1942_mult__pos__neg2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),zero_z2226904508553997617atural) ) ) ).

% mult_pos_neg2
tff(fact_1943_mult__pos__neg2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),zero_z3403309356797280102nteger) ) ) ).

% mult_pos_neg2
tff(fact_1944_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),zero_zero_rat) ) ) ).

% mult_pos_neg2
tff(fact_1945_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,B),A)),zero_zero_nat) ) ) ).

% mult_pos_neg2
tff(fact_1946_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),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,B),A)),zero_zero_int) ) ) ).

% mult_pos_neg2
tff(fact_1947_zero__less__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) )
        | ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger) ) ) ) ).

% zero_less_mult_iff
tff(fact_1948_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_rat(zero_zero_rat),B) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_rat(B),zero_zero_rat) ) ) ) ).

% zero_less_mult_iff
tff(fact_1949_zero__less__mult__iff,axiom,
    ! [A: int,B: 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,A),B))
    <=> ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) )
        | ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
          & aa_int_o(aa_int_fun_int_o(ord_less_int,B),zero_zero_int) ) ) ) ).

% zero_less_mult_iff
tff(fact_1950_zero__less__mult__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B))
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B) ) ) ).

% zero_less_mult_pos
tff(fact_1951_zero__less__mult__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos
tff(fact_1952_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
       => aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1953_zero__less__mult__pos,axiom,
    ! [A: nat,B: 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,A),B))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos
tff(fact_1954_zero__less__mult__pos,axiom,
    ! [A: int,B: 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,A),B))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos
tff(fact_1955_zero__less__mult__pos2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A))
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1956_zero__less__mult__pos2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A))
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1957_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A))
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
       => aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1958_zero__less__mult__pos2,axiom,
    ! [B: nat,A: 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,B),A))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1959_zero__less__mult__pos2,axiom,
    ! [B: int,A: 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,B),A))
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ) ).

% zero_less_mult_pos2
tff(fact_1960_mult__less__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1961_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(ord_less_rat(B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1962_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1963_mult__less__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1964_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(ord_less_rat(A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1965_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1966_mult__strict__left__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1967_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1968_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1969_mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1970_mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1971_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1972_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1973_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% mult_strict_left_mono
tff(fact_1974_mult__less__cancel__left__disj,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) )
        | ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1975_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
          & aa_rat_o(ord_less_rat(A),B) )
        | ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
          & aa_rat_o(ord_less_rat(B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1976_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( 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,A),B) )
        | ( 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,B),A) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1977_mult__strict__right__mono__neg,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1978_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1979_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1980_mult__strict__right__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1981_mult__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1982_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1983_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1984_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% mult_strict_right_mono
tff(fact_1985_mult__less__cancel__right__disj,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) )
        | ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
          & aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1986_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
          & aa_rat_o(ord_less_rat(A),B) )
        | ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
          & aa_rat_o(ord_less_rat(B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1987_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B) )
        | ( 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,B),A) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1988_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1989_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1990_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1991_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1992_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1993_less__numeral__extra_I1_J,axiom,
    aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),one_one_Code_natural) ).

% less_numeral_extra(1)
tff(fact_1994_less__numeral__extra_I1_J,axiom,
    aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),one_one_Code_integer) ).

% less_numeral_extra(1)
tff(fact_1995_less__numeral__extra_I1_J,axiom,
    aa_rat_o(ord_less_rat(zero_zero_rat),one_one_rat) ).

% less_numeral_extra(1)
tff(fact_1996_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_1997_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_1998_zero__less__one,axiom,
    aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),one_one_Code_natural) ).

% zero_less_one
tff(fact_1999_zero__less__one,axiom,
    aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),one_one_Code_integer) ).

% zero_less_one
tff(fact_2000_zero__less__one,axiom,
    aa_rat_o(ord_less_rat(zero_zero_rat),one_one_rat) ).

% zero_less_one
tff(fact_2001_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_2002_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_2003_not__one__less__zero,axiom,
    ~ aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),zero_z2226904508553997617atural) ).

% not_one_less_zero
tff(fact_2004_not__one__less__zero,axiom,
    ~ aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),zero_z3403309356797280102nteger) ).

% not_one_less_zero
tff(fact_2005_not__one__less__zero,axiom,
    ~ aa_rat_o(ord_less_rat(one_one_rat),zero_zero_rat) ).

% not_one_less_zero
tff(fact_2006_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_2007_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_2008_less__1__mult,axiom,
    ! [M2: code_natural,N: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),M2)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),N)
       => aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,M2),N)) ) ) ).

% less_1_mult
tff(fact_2009_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),M2)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),N)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,M2),N)) ) ) ).

% less_1_mult
tff(fact_2010_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),M2)
     => ( aa_rat_o(ord_less_rat(one_one_rat),N)
       => aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,M2),N)) ) ) ).

% less_1_mult
tff(fact_2011_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
     => ( 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,M2),N)) ) ) ).

% less_1_mult
tff(fact_2012_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),M2)
     => ( 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,M2),N)) ) ) ).

% less_1_mult
tff(fact_2013_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_2014_less__minus__one__simps_I4_J,axiom,
    ~ aa_rat_o(ord_less_rat(one_one_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(4)
tff(fact_2015_less__minus__one__simps_I4_J,axiom,
    ~ aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(4)
tff(fact_2016_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_2017_less__minus__one__simps_I2_J,axiom,
    aa_rat_o(ord_less_rat(uminus_uminus_rat(one_one_rat)),one_one_rat) ).

% less_minus_one_simps(2)
tff(fact_2018_less__minus__one__simps_I2_J,axiom,
    aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) ).

% less_minus_one_simps(2)
tff(fact_2019_splice_Oelims,axiom,
    ! [X: list_P8527749157015355191n_assn,Xa: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( splice4588524007259616650n_assn(X,Xa) = Y )
     => ( ( ( X = nil_Pr5671120429643327159n_assn )
         => ( Y != Xa ) )
       => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( X = cons_P2971678138204555879n_assn(X2,Xs2) )
             => ( Y != cons_P2971678138204555879n_assn(X2,splice4588524007259616650n_assn(Xa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_2020_splice_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( splice_int(X,Xa) = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ~ ! [X2: int,Xs2: list_int] :
              ( ( X = cons_int(X2,Xs2) )
             => ( Y != cons_int(X2,splice_int(Xa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_2021_splice_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( splice_nat(X,Xa) = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ~ ! [X2: nat,Xs2: list_nat] :
              ( ( X = cons_nat(X2,Xs2) )
             => ( Y != cons_nat(X2,splice_nat(Xa,Xs2)) ) ) ) ) ).

% splice.elims
tff(fact_2022_measures__lesseq,axiom,
    ! [F: fun_nat_nat,X: nat,Y: nat,Fs: list_fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(F,X)),aa_nat_nat(F,Y))
     => ( member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(Fs))
       => member8440522571783428010at_nat(product_Pair_nat_nat(X,Y),measures_nat(cons_fun_nat_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2023_measures__lesseq,axiom,
    ! [F: fun_int_nat,X: int,Y: int,Fs: list_fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_int_nat(F,X)),aa_int_nat(F,Y))
     => ( member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(Fs))
       => member5262025264175285858nt_int(product_Pair_int_int(X,Y),measures_int(cons_fun_int_nat(F,Fs))) ) ) ).

% measures_lesseq
tff(fact_2024_mult__less__le__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(C),D2)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2025_mult__less__le__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2026_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),D2)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
         => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2027_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2028_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_2029_mult__le__less__imp__less,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(C),D2)
       => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2030_mult__le__less__imp__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2)
       => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2031_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),D2)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2032_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2033_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_2034_mult__right__le__imp__le,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2035_mult__right__le__imp__le,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2036_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_eq_rat(A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2037_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2038_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B) ) ) ).

% mult_right_le_imp_le
tff(fact_2039_mult__left__le__imp__le,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2040_mult__left__le__imp__le,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2041_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_eq_rat(A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2042_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( 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,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2043_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( 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,A),B) ) ) ).

% mult_left_le_imp_le
tff(fact_2044_mult__le__cancel__left__pos,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2045_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2046_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% mult_le_cancel_left_pos
tff(fact_2047_mult__le__cancel__left__neg,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(ord_le3102999989581377725nteger(B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2048_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
      <=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2049_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% mult_le_cancel_left_neg
tff(fact_2050_mult__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2051_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(A),B) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2052_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B) )
        & ( 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,B),A) ) ) ) ).

% mult_less_cancel_right
tff(fact_2053_mult__strict__mono_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(C),D2)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
         => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2054_mult__strict__mono_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2055_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),D2)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2056_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2057_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono'
tff(fact_2058_mult__right__less__imp__less,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2059_mult__right__less__imp__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2060_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_rat(A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2061_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2062_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B) ) ) ).

% mult_right_less_imp_less
tff(fact_2063_mult__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2064_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(A),B) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2065_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( 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,A),B) )
        & ( 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,B),A) ) ) ) ).

% mult_less_cancel_left
tff(fact_2066_mult__strict__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(C),D2)
       => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
         => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
           => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2067_mult__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2)
       => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
           => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2068_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),D2)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
           => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2069_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
         => ( 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2070_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
         => ( 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ) ).

% mult_strict_mono
tff(fact_2071_mult__left__less__imp__less,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2072_mult__left__less__imp__less,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2073_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => aa_rat_o(ord_less_rat(A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2074_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
     => ( 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,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2075_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
     => ( 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,A),B) ) ) ).

% mult_left_less_imp_less
tff(fact_2076_mult__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2077_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(A),B) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2078_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B) )
        & ( 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,B),A) ) ) ) ).

% mult_le_cancel_right
tff(fact_2079_mult__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2080_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(A),B) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2081_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( 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,A),B) )
        & ( 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,B),A) ) ) ) ).

% mult_le_cancel_left
tff(fact_2082_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_2083_less__minus__one__simps_I3_J,axiom,
    ~ aa_rat_o(ord_less_rat(zero_zero_rat),uminus_uminus_rat(one_one_rat)) ).

% less_minus_one_simps(3)
tff(fact_2084_less__minus__one__simps_I3_J,axiom,
    ~ aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% less_minus_one_simps(3)
tff(fact_2085_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_2086_less__minus__one__simps_I1_J,axiom,
    aa_rat_o(ord_less_rat(uminus_uminus_rat(one_one_rat)),zero_zero_rat) ).

% less_minus_one_simps(1)
tff(fact_2087_less__minus__one__simps_I1_J,axiom,
    aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),zero_z3403309356797280102nteger) ).

% less_minus_one_simps(1)
tff(fact_2088_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(A),one_one_Code_integer) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2089_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(A),one_one_rat) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(one_one_rat),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2090_mult__less__cancel__right2,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% mult_less_cancel_right2
tff(fact_2091_mult__less__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),B) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2092_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(one_one_rat),B) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(B),one_one_rat) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2093_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(times_times_int,B),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),B) )
        & ( 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,B),one_one_int) ) ) ) ).

% mult_less_cancel_right1
tff(fact_2094_mult__less__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(A),one_one_Code_integer) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2095_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(A),one_one_rat) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(one_one_rat),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2096_mult__less__cancel__left2,axiom,
    ! [C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),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,A),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),A) ) ) ) ).

% mult_less_cancel_left2
tff(fact_2097_mult__less__cancel__left1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),B) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(B),one_one_Code_integer) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2098_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_rat(one_one_rat),B) )
        & ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_rat(B),one_one_rat) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2099_mult__less__cancel__left1,axiom,
    ! [C: int,B: 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),B))
    <=> ( ( 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),B) )
        & ( 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,B),one_one_int) ) ) ) ).

% mult_less_cancel_left1
tff(fact_2100_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),C)
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2101_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),C)
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(A),one_one_rat) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(one_one_rat),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2102_mult__le__cancel__right2,axiom,
    ! [A: 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,A),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,A),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),A) ) ) ) ).

% mult_le_cancel_right2
tff(fact_2103_mult__le__cancel__right1,axiom,
    ! [C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),B) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(B),one_one_Code_integer) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2104_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(one_one_rat),B) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(B),one_one_rat) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2105_mult__le__cancel__right1,axiom,
    ! [C: int,B: 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,B),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),B) )
        & ( 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,B),one_one_int) ) ) ) ).

% mult_le_cancel_right1
tff(fact_2106_mult__le__cancel__left2,axiom,
    ! [C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),C)
    <=> ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),C)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer) )
        & ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),zero_z3403309356797280102nteger)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2107_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),C)
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
         => aa_rat_o(ord_less_eq_rat(A),one_one_rat) )
        & ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
         => aa_rat_o(ord_less_eq_rat(one_one_rat),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2108_mult__le__cancel__left2,axiom,
    ! [C: int,A: 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),A)),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,A),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),A) ) ) ) ).

% mult_le_cancel_left2
tff(fact_2109_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( aa_rat_o(ord_less_rat(zero_zero_rat),Z2)
         => ( aa_rat_o(ord_less_rat(Z2),one_one_rat)
           => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z2),X)),Y) ) )
     => aa_rat_o(ord_less_eq_rat(X),Y) ) ).

% field_le_mult_one_interval
tff(fact_2110_mult__le__cancel__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(ord_le3102999989581377725nteger(X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2111_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(ord_less_eq_rat(X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2112_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: 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,X),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,X),Y) ) ) ).

% mult_le_cancel_iff1
tff(fact_2113_mult__le__cancel__iff2,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),Y))
      <=> aa_Code_integer_o(ord_le3102999989581377725nteger(X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2114_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
      <=> aa_rat_o(ord_less_eq_rat(X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2115_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: 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),X)),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,X),Y) ) ) ).

% mult_le_cancel_iff2
tff(fact_2116_mult__less__iff1,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),Z)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Z))
      <=> aa_Code_integer_o(ord_le6747313008572928689nteger(X),Y) ) ) ).

% mult_less_iff1
tff(fact_2117_mult__less__iff1,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Z)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))
      <=> aa_rat_o(ord_less_rat(X),Y) ) ) ).

% mult_less_iff1
tff(fact_2118_mult__less__iff1,axiom,
    ! [Z: int,X: 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,X),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,X),Y) ) ) ).

% mult_less_iff1
tff(fact_2119_quicksort_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ~ ! [X2: int,Xs2: list_int] : X != cons_int(X2,Xs2) ) ).

% quicksort.cases
tff(fact_2120_quicksort_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] : X != cons_nat(X2,Xs2) ) ).

% quicksort.cases
tff(fact_2121_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc337509097454979815n_assn] :
      ( ! [Ys3: list_P8527749157015355191n_assn] : X != produc3014731801419794263n_assn(nil_Pr5671120429643327159n_assn,Ys3)
     => ~ ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] : X != produc3014731801419794263n_assn(cons_P2971678138204555879n_assn(X2,Xs2),Ys3) ) ).

% subset_eq_mset_impl.cases
tff(fact_2122_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc1186641810826059865st_int] :
      ( ! [Ys3: list_int] : X != produc364263696895485585st_int(nil_int,Ys3)
     => ~ ! [X2: int,Xs2: list_int,Ys3: list_int] : X != produc364263696895485585st_int(cons_int(X2,Xs2),Ys3) ) ).

% subset_eq_mset_impl.cases
tff(fact_2123_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] : X != produc2694037385005941721st_nat(nil_nat,Ys3)
     => ~ ! [X2: nat,Xs2: list_nat,Ys3: list_nat] : X != produc2694037385005941721st_nat(cons_nat(X2,Xs2),Ys3) ) ).

% subset_eq_mset_impl.cases
tff(fact_2124_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4918476307565957903at_nat(inf_in2572325071724192079at_nat,top_to4669805908274784177at_nat,ord_le3146513528884898305at_nat,ord_le7866589430770878221at_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_2125_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4431159887699967284_int_o(inf_in1782598092406300404_int_o,top_to8487252020945707026_int_o,ord_le14110778710936002_int_o,ord_le7515035663206590902_int_o) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_2126_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila4423693302617054363t_unit(inf_in4660618365625256667t_unit,top_to1996260823553986621t_unit,ord_le3507040750410214029t_unit,ord_le8056459307392131481t_unit) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_2127_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila8603258263270017530r_assn(inf_inf_assn,top_top_assn,ord_less_eq_assn,ord_less_assn) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_2128_inf__top_Osemilattice__neutr__order__axioms,axiom,
    semila1667268886620078168et_nat(inf_inf_set_nat,top_top_set_nat,ord_less_eq_set_nat,ord_less_set_nat) ).

% inf_top.semilattice_neutr_order_axioms
tff(fact_2129_sgn__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% sgn_neg
tff(fact_2130_sgn__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) ) ) ).

% sgn_neg
tff(fact_2131_sgn__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% sgn_neg
tff(fact_2132_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] : insert3246601298802261197n_assn(X,nil_Pr5671120429643327159n_assn) = cons_P2971678138204555879n_assn(X,nil_Pr5671120429643327159n_assn) ).

% insert_Nil
tff(fact_2133_insert__Nil,axiom,
    ! [X: int] : insert_int(X,nil_int) = cons_int(X,nil_int) ).

% insert_Nil
tff(fact_2134_insert__Nil,axiom,
    ! [X: nat] : insert_nat(X,nil_nat) = cons_nat(X,nil_nat) ).

% insert_Nil
tff(fact_2135_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(X),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(Y),A)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),V2)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V2) = one_one_Code_integer )
             => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2136_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(ord_less_rat(X),A)
     => ( aa_rat_o(ord_less_rat(Y),A)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),U)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),V2)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2137_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,Y),A)
       => ( 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),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_2138_add__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2139_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2140_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2141_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add_left_cancel
tff(fact_2142_add__right__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2143_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2144_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2145_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add_right_cancel
tff(fact_2146_add__le__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% add_le_cancel_left
tff(fact_2147_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(ord_less_eq_rat(A),B) ) ).

% add_le_cancel_left
tff(fact_2148_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_left
tff(fact_2149_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_left
tff(fact_2150_add__le__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% add_le_cancel_right
tff(fact_2151_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(ord_less_eq_rat(A),B) ) ).

% add_le_cancel_right
tff(fact_2152_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_cancel_right
tff(fact_2153_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_cancel_right
tff(fact_2154_add_Oright__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.right_neutral
tff(fact_2155_add_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.right_neutral
tff(fact_2156_add_Oright__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.right_neutral
tff(fact_2157_add_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.right_neutral
tff(fact_2158_add_Oright__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).

% add.right_neutral
tff(fact_2159_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A) )
    <=> ( A = zero_zero_int ) ) ).

% double_zero_sym
tff(fact_2160_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A) )
    <=> ( A = zero_zero_rat ) ) ).

% double_zero_sym
tff(fact_2161_double__zero__sym,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% double_zero_sym
tff(fact_2162_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
tff(fact_2163_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_left
tff(fact_2164_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
tff(fact_2165_add__cancel__left__left,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_left
tff(fact_2166_add__cancel__left__left,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) = A )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_left
tff(fact_2167_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = A )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
tff(fact_2168_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = A )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_left_right
tff(fact_2169_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = A )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
tff(fact_2170_add__cancel__left__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = A )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_left_right
tff(fact_2171_add__cancel__left__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) = A )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_left_right
tff(fact_2172_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
tff(fact_2173_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_left
tff(fact_2174_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
tff(fact_2175_add__cancel__right__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_left
tff(fact_2176_add__cancel__right__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A) )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_left
tff(fact_2177_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) )
    <=> ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
tff(fact_2178_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) )
    <=> ( B = zero_zero_int ) ) ).

% add_cancel_right_right
tff(fact_2179_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) )
    <=> ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
tff(fact_2180_add__cancel__right__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) )
    <=> ( B = zero_z3403309356797280102nteger ) ) ).

% add_cancel_right_right
tff(fact_2181_add__cancel__right__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B) )
    <=> ( B = zero_z2226904508553997617atural ) ) ).

% add_cancel_right_right
tff(fact_2182_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) = zero_zero_nat )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_2183_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y) )
    <=> ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_2184_add__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% add_0
tff(fact_2185_add__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add_0
tff(fact_2186_add__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add_0
tff(fact_2187_add__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add_0
tff(fact_2188_add__0,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).

% add_0
tff(fact_2189_add__less__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ).

% add_less_cancel_left
tff(fact_2190_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
    <=> aa_rat_o(ord_less_rat(A),B) ) ).

% add_less_cancel_left
tff(fact_2191_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_left
tff(fact_2192_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_left
tff(fact_2193_add__less__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ).

% add_less_cancel_right
tff(fact_2194_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
    <=> aa_rat_o(ord_less_rat(A),B) ) ).

% add_less_cancel_right
tff(fact_2195_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_cancel_right
tff(fact_2196_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_cancel_right
tff(fact_2197_add__minus__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B)) = B ).

% add_minus_cancel
tff(fact_2198_add__minus__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B)) = B ).

% add_minus_cancel
tff(fact_2199_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)) = B ).

% add_minus_cancel
tff(fact_2200_minus__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = B ).

% minus_add_cancel
tff(fact_2201_minus__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = B ).

% minus_add_cancel
tff(fact_2202_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = B ).

% minus_add_cancel
tff(fact_2203_minus__add__distrib,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(uminus_uminus_int,B)) ).

% minus_add_distrib
tff(fact_2204_minus__add__distrib,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),uminus_uminus_rat(B)) ).

% minus_add_distrib
tff(fact_2205_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% minus_add_distrib
tff(fact_2206_sgn__1,axiom,
    aa_int_int(sgn_sgn_int,one_one_int) = one_one_int ).

% sgn_1
tff(fact_2207_sgn__1,axiom,
    sgn_sgn_rat(one_one_rat) = one_one_rat ).

% sgn_1
tff(fact_2208_sgn__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,one_one_Code_integer) = one_one_Code_integer ).

% sgn_1
tff(fact_2209_add__le__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel1
tff(fact_2210_add__le__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B)
    <=> aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural) ) ).

% add_le_same_cancel1
tff(fact_2211_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% add_le_same_cancel1
tff(fact_2212_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel1
tff(fact_2213_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel1
tff(fact_2214_add__le__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) ) ).

% add_le_same_cancel2
tff(fact_2215_add__le__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B)
    <=> aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural) ) ).

% add_le_same_cancel2
tff(fact_2216_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% add_le_same_cancel2
tff(fact_2217_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat) ) ).

% add_le_same_cancel2
tff(fact_2218_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% add_le_same_cancel2
tff(fact_2219_le__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel1
tff(fact_2220_le__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B) ) ).

% le_add_same_cancel1
tff(fact_2221_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ).

% le_add_same_cancel1
tff(fact_2222_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel1
tff(fact_2223_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel1
tff(fact_2224_le__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B) ) ).

% le_add_same_cancel2
tff(fact_2225_le__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A))
    <=> aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B) ) ).

% le_add_same_cancel2
tff(fact_2226_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),B) ) ).

% le_add_same_cancel2
tff(fact_2227_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B) ) ).

% le_add_same_cancel2
tff(fact_2228_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B) ) ).

% le_add_same_cancel2
tff(fact_2229_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2230_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2231_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_2232_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2233_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2234_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: 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,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_2235_add__less__same__cancel1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A)),B)
    <=> aa_Code_natural_o(ord_le5570908160329646204atural(A),zero_z2226904508553997617atural) ) ).

% add_less_same_cancel1
tff(fact_2236_add__less__same__cancel1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A)),B)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel1
tff(fact_2237_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A)),B)
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% add_less_same_cancel1
tff(fact_2238_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel1
tff(fact_2239_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel1
tff(fact_2240_add__less__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),B)
    <=> aa_Code_natural_o(ord_le5570908160329646204atural(A),zero_z2226904508553997617atural) ) ).

% add_less_same_cancel2
tff(fact_2241_add__less__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),B)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% add_less_same_cancel2
tff(fact_2242_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B)
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% add_less_same_cancel2
tff(fact_2243_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat) ) ).

% add_less_same_cancel2
tff(fact_2244_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% add_less_same_cancel2
tff(fact_2245_less__add__same__cancel1,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B))
    <=> aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B) ) ).

% less_add_same_cancel1
tff(fact_2246_less__add__same__cancel1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel1
tff(fact_2247_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ).

% less_add_same_cancel1
tff(fact_2248_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel1
tff(fact_2249_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel1
tff(fact_2250_less__add__same__cancel2,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A))
    <=> aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B) ) ).

% less_add_same_cancel2
tff(fact_2251_less__add__same__cancel2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B) ) ).

% less_add_same_cancel2
tff(fact_2252_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A))
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),B) ) ).

% less_add_same_cancel2
tff(fact_2253_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B) ) ).

% less_add_same_cancel2
tff(fact_2254_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B) ) ).

% less_add_same_cancel2
tff(fact_2255_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A)),zero_z3403309356797280102nteger)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2256_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2257_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),A)),zero_zero_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_2258_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),A))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2259_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),A))
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2260_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: 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,A),A))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_2261_add_Oright__inverse,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,A)) = zero_zero_int ).

% add.right_inverse
tff(fact_2262_add_Oright__inverse,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(A)) = zero_zero_rat ).

% add.right_inverse
tff(fact_2263_add_Oright__inverse,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = zero_z3403309356797280102nteger ).

% add.right_inverse
tff(fact_2264_ab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_left_minus
tff(fact_2265_ab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_left_minus
tff(fact_2266_ab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_left_minus
tff(fact_2267_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_2268_add__neg__numeral__special_I7_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% add_neg_numeral_special(7)
tff(fact_2269_add__neg__numeral__special_I7_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(7)
tff(fact_2270_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_2271_add__neg__numeral__special_I8_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) = zero_zero_rat ).

% add_neg_numeral_special(8)
tff(fact_2272_add__neg__numeral__special_I8_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% add_neg_numeral_special(8)
tff(fact_2273_sgn__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer ) ) ).

% sgn_pos
tff(fact_2274_sgn__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( sgn_sgn_rat(A) = one_one_rat ) ) ).

% sgn_pos
tff(fact_2275_sgn__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(sgn_sgn_int,A) = one_one_int ) ) ).

% sgn_pos
tff(fact_2276_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),X)
    <=> ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
tff(fact_2277_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2278_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2279_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2280_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L) ) ) ).

% add_mono_thms_linordered_semiring(4)
tff(fact_2281_group__cancel_Oadd1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A2),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2282_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2283_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),A) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A2),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2284_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add1
tff(fact_2285_group__cancel_Oadd2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2286_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2287_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2 = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),B) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B2) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,K),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2288_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) ) ) ).

% group_cancel.add2
tff(fact_2289_add_Oright__assoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% add.right_assoc
tff(fact_2290_add_Oright__assoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.right_assoc
tff(fact_2291_add_Oright__assoc,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.right_assoc
tff(fact_2292_add_Oright__assoc,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.right_assoc
tff(fact_2293_add_Oright__commute,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ).

% add.right_commute
tff(fact_2294_add_Oright__commute,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% add.right_commute
tff(fact_2295_add_Oright__commute,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ).

% add.right_commute
tff(fact_2296_add_Oright__commute,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% add.right_commute
tff(fact_2297_add_Oassoc,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% add.assoc
tff(fact_2298_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% add.assoc
tff(fact_2299_add_Oassoc,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% add.assoc
tff(fact_2300_add_Oassoc,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% add.assoc
tff(fact_2301_add_Oleft__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2302_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2303_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
    <=> ( B = C ) ) ).

% add.left_cancel
tff(fact_2304_add_Oright__cancel,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2305_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2306_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
    <=> ( B = C ) ) ).

% add.right_cancel
tff(fact_2307_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_2308_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_2309_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_2310_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ).

% ab_semigroup_add_class.add.commute
tff(fact_2311_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2312_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2313_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2314_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_2315_add__left__imp__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2316_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2317_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2318_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) )
     => ( B = C ) ) ).

% add_left_imp_eq
tff(fact_2319_add__right__imp__eq,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2320_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2321_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2322_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A) )
     => ( B = C ) ) ).

% add_right_imp_eq
tff(fact_2323_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(I2),J)
        & ( K = L ) )
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2324_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_eq_rat(I2),J)
        & ( K = L ) )
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2325_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2326_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_2327_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & aa_Code_integer_o(ord_le3102999989581377725nteger(K),L) )
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2328_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(ord_less_eq_rat(K),L) )
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2329_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = 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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2330_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = 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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_2331_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(I2),J)
        & aa_Code_integer_o(ord_le3102999989581377725nteger(K),L) )
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2332_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_eq_rat(I2),J)
        & aa_rat_o(ord_less_eq_rat(K),L) )
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2333_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2334_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_2335_add__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_mono
tff(fact_2336_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),D2)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_mono
tff(fact_2337_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_mono
tff(fact_2338_add__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_mono
tff(fact_2339_add__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% add_left_mono
tff(fact_2340_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_left_mono
tff(fact_2341_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_left_mono
tff(fact_2342_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_left_mono
tff(fact_2343_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ~ ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) ) ).

% less_eqE
tff(fact_2344_add__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).

% add_right_mono
tff(fact_2345_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_right_mono
tff(fact_2346_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_right_mono
tff(fact_2347_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_right_mono
tff(fact_2348_le__iff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ? [C4: nat] : B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C4) ) ).

% le_iff_add
tff(fact_2349_add__le__imp__le__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
     => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% add_le_imp_le_left
tff(fact_2350_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(ord_less_eq_rat(A),B) ) ).

% add_le_imp_le_left
tff(fact_2351_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_left
tff(fact_2352_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_left
tff(fact_2353_add__le__imp__le__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
     => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% add_le_imp_le_right
tff(fact_2354_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(ord_less_eq_rat(A),B) ) ).

% add_le_imp_le_right
tff(fact_2355_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B) ) ).

% add_le_imp_le_right
tff(fact_2356_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% add_le_imp_le_right
tff(fact_2357_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_zero_nat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2358_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2359_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2360_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2361_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_z2226904508553997617atural),A) = A ).

% comm_monoid_add_class.add_0
tff(fact_2362_add_Ocomm__neutral,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_zero_nat) = A ).

% add.comm_neutral
tff(fact_2363_add_Ocomm__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_zero_int) = A ).

% add.comm_neutral
tff(fact_2364_add_Ocomm__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),zero_zero_rat) = A ).

% add.comm_neutral
tff(fact_2365_add_Ocomm__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_z3403309356797280102nteger) = A ).

% add.comm_neutral
tff(fact_2366_add_Ocomm__neutral,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_z2226904508553997617atural) = A ).

% add.comm_neutral
tff(fact_2367_add_Ogroup__left__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_zero_int),A) = A ).

% add.group_left_neutral
tff(fact_2368_add_Ogroup__left__neutral,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,zero_zero_rat),A) = A ).

% add.group_left_neutral
tff(fact_2369_add_Ogroup__left__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_z3403309356797280102nteger),A) = A ).

% add.group_left_neutral
tff(fact_2370_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(I2),J)
        & aa_Code_integer_o(ord_le6747313008572928689nteger(K),L) )
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2371_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_rat(I2),J)
        & aa_rat_o(ord_less_rat(K),L) )
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2372_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2373_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_2374_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( ( I2 = J )
        & aa_Code_integer_o(ord_le6747313008572928689nteger(K),L) )
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2375_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & aa_rat_o(ord_less_rat(K),L) )
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2376_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = 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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2377_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = 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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_2378_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(I2),J)
        & ( K = L ) )
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2379_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_rat(I2),J)
        & ( K = L ) )
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2380_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2381_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_2382_add__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2383_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),D2)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2384_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2385_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_strict_mono
tff(fact_2386_add__strict__left__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% add_strict_left_mono
tff(fact_2387_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% add_strict_left_mono
tff(fact_2388_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) ) ).

% add_strict_left_mono
tff(fact_2389_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% add_strict_left_mono
tff(fact_2390_add__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ) ).

% add_strict_right_mono
tff(fact_2391_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ) ).

% add_strict_right_mono
tff(fact_2392_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ) ).

% add_strict_right_mono
tff(fact_2393_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ) ).

% add_strict_right_mono
tff(fact_2394_add__less__imp__less__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B))
     => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ).

% add_less_imp_less_left
tff(fact_2395_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B))
     => aa_rat_o(ord_less_rat(A),B) ) ).

% add_less_imp_less_left
tff(fact_2396_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_left
tff(fact_2397_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_left
tff(fact_2398_add__less__imp__less__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C))
     => aa_Code_integer_o(ord_le6747313008572928689nteger(A),B) ) ).

% add_less_imp_less_right
tff(fact_2399_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C))
     => aa_rat_o(ord_less_rat(A),B) ) ).

% add_less_imp_less_right
tff(fact_2400_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C))
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B) ) ).

% add_less_imp_less_right
tff(fact_2401_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C))
     => aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ).

% add_less_imp_less_right
tff(fact_2402_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2403_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: 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,A),E2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),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,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2404_combine__common__factor,axiom,
    ! [A: int,E2: int,B: 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,A),E2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),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,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2405_combine__common__factor,axiom,
    ! [A: code_integer,E2: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2406_combine__common__factor,axiom,
    ! [A: code_natural,E2: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),E2)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),E2)),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),E2)),C) ).

% combine_common_factor
tff(fact_2407_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% distrib_right
tff(fact_2408_distrib__right,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% distrib_right
tff(fact_2409_distrib__right,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% distrib_right
tff(fact_2410_distrib__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% distrib_right
tff(fact_2411_distrib__right,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ).

% distrib_right
tff(fact_2412_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% distrib_left
tff(fact_2413_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% distrib_left
tff(fact_2414_distrib__left,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% distrib_left
tff(fact_2415_distrib__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% distrib_left
tff(fact_2416_distrib__left,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)) ).

% distrib_left
tff(fact_2417_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2418_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2419_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2420_comm__semiring__class_Odistrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2421_comm__semiring__class_Odistrib,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ).

% comm_semiring_class.distrib
tff(fact_2422_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2423_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2424_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% ring_class.ring_distribs(1)
tff(fact_2425_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2426_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2427_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% ring_class.ring_distribs(2)
tff(fact_2428_group__cancel_Oneg1,axiom,
    ! [A2: int,K: int,A: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(uminus_uminus_int,A2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(uminus_uminus_int,A)) ) ) ).

% group_cancel.neg1
tff(fact_2429_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( uminus_uminus_rat(A2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K)),uminus_uminus_rat(A)) ) ) ).

% group_cancel.neg1
tff(fact_2430_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ) ) ).

% group_cancel.neg1
tff(fact_2431_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,B)),aa_int_int(uminus_uminus_int,A)) ).

% add.inverse_distrib_swap
tff(fact_2432_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(B)),uminus_uminus_rat(A)) ).

% add.inverse_distrib_swap
tff(fact_2433_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) ).

% add.inverse_distrib_swap
tff(fact_2434_sgn__mult,axiom,
    ! [A: rat,B: rat] : sgn_sgn_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(B)) ).

% sgn_mult
tff(fact_2435_sgn__mult,axiom,
    ! [A: int,B: int] : aa_int_int(sgn_sgn_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,B)) ).

% sgn_mult
tff(fact_2436_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,B)) ).

% sgn_mult
tff(fact_2437_add_Oabel__semigroup__axioms,axiom,
    abel_s8772020223019846471nteger(plus_p5714425477246183910nteger) ).

% add.abel_semigroup_axioms
tff(fact_2438_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_rat(plus_plus_rat) ).

% add.abel_semigroup_axioms
tff(fact_2439_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_nat(plus_plus_nat) ).

% add.abel_semigroup_axioms
tff(fact_2440_add_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int(plus_plus_int) ).

% add.abel_semigroup_axioms
tff(fact_2441_add_Osemigroup__axioms,axiom,
    semigr7036097719740861967nteger(plus_p5714425477246183910nteger) ).

% add.semigroup_axioms
tff(fact_2442_add_Osemigroup__axioms,axiom,
    semigroup_rat(plus_plus_rat) ).

% add.semigroup_axioms
tff(fact_2443_add_Osemigroup__axioms,axiom,
    semigroup_nat(plus_plus_nat) ).

% add.semigroup_axioms
tff(fact_2444_add_Osemigroup__axioms,axiom,
    semigroup_int(plus_plus_int) ).

% add.semigroup_axioms
tff(fact_2445_add_Oac__operator__axioms,axiom,
    syntax414191934074104506nteger(plus_p5714425477246183910nteger) ).

% add.ac_operator_axioms
tff(fact_2446_add_Oac__operator__axioms,axiom,
    syntax8165409007564224565or_rat(plus_plus_rat) ).

% add.ac_operator_axioms
tff(fact_2447_add_Oac__operator__axioms,axiom,
    syntax8800539067650720301or_nat(plus_plus_nat) ).

% add.ac_operator_axioms
tff(fact_2448_add_Oac__operator__axioms,axiom,
    syntax8798048597141670025or_int(plus_plus_int) ).

% add.ac_operator_axioms
tff(fact_2449_add_Osafe__commute,axiom,
    ! [X: code_integer,Y: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(syntax380677951125133566nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A) ) ) ).

% add.safe_commute
tff(fact_2450_add_Osafe__commute,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( aa_rat_o(syntax3730441303064801268at_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),A) ) ) ).

% add.safe_commute
tff(fact_2451_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(syntax4682126007086162916at_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),A) ) ) ).

% add.safe_commute
tff(fact_2452_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(syntax5678989248478167196nt_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,B),A) ) ) ).

% add.safe_commute
tff(fact_2453_add__decreasing,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2454_add__decreasing,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(C),B)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2455_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(C),B)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2456_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2457_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing
tff(fact_2458_add__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing
tff(fact_2459_add__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),C)
       => aa_Code_natural_o(ord_le1926595141338095240atural(B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_increasing
tff(fact_2460_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(B),C)
       => aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing
tff(fact_2461_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing
tff(fact_2462_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing
tff(fact_2463_add__decreasing2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2464_add__decreasing2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(C),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(A),B)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2465_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(A),B)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2466_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: 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,A),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2467_add__decreasing2,axiom,
    ! [C: int,A: int,B: 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,A),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ) ) ).

% add_decreasing2
tff(fact_2468_add__increasing2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),A)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_increasing2
tff(fact_2469_add__increasing2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),C)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),A)
       => aa_Code_natural_o(ord_le1926595141338095240atural(B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_increasing2
tff(fact_2470_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(B),A)
       => aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_increasing2
tff(fact_2471_add__increasing2,axiom,
    ! [C: nat,B: nat,A: 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,B),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_increasing2
tff(fact_2472_add__increasing2,axiom,
    ! [C: int,B: int,A: 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,B),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_increasing2
tff(fact_2473_add__nonneg__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2474_add__nonneg__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2475_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2476_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2477_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% add_nonneg_nonneg
tff(fact_2478_add__nonpos__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_nonpos
tff(fact_2479_add__nonpos__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le1926595141338095240atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_nonpos_nonpos
tff(fact_2480_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_nonpos
tff(fact_2481_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_nonpos
tff(fact_2482_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)),zero_zero_int) ) ) ).

% add_nonpos_nonpos
tff(fact_2483_add__nonneg__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),X)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),Y)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2484_add__nonneg__eq__0__iff,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),X)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),Y)
       => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y) = zero_z2226904508553997617atural )
        <=> ( ( X = zero_z2226904508553997617atural )
            & ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2485_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),Y)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2486_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
     => ( 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,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2487_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( 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,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_2488_add__nonpos__eq__0__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(X),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(Y),zero_z3403309356797280102nteger)
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y) = zero_z3403309356797280102nteger )
        <=> ( ( X = zero_z3403309356797280102nteger )
            & ( Y = zero_z3403309356797280102nteger ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2489_add__nonpos__eq__0__iff,axiom,
    ! [X: code_natural,Y: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(X),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(Y),zero_z2226904508553997617atural)
       => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y) = zero_z2226904508553997617atural )
        <=> ( ( X = zero_z2226904508553997617atural )
            & ( Y = zero_z2226904508553997617atural ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2490_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(Y),zero_zero_rat)
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y) = zero_zero_rat )
        <=> ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2491_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),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,X),Y) = zero_zero_nat )
        <=> ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2492_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),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,X),Y) = zero_zero_int )
        <=> ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_2493_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(I2),J)
        & aa_Code_integer_o(ord_le6747313008572928689nteger(K),L) )
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2494_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_eq_rat(I2),J)
        & aa_rat_o(ord_less_rat(K),L) )
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2495_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2496_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_2497_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: code_integer,J: code_integer,K: code_integer,L: code_integer] :
      ( ( aa_Code_integer_o(ord_le6747313008572928689nteger(I2),J)
        & aa_Code_integer_o(ord_le3102999989581377725nteger(K),L) )
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,I2),K)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2498_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( aa_rat_o(ord_less_rat(I2),J)
        & aa_rat_o(ord_less_eq_rat(K),L) )
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,I2),K)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2499_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),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,I2),K)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2500_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),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,I2),K)),aa_int_int(aa_int_fun_int_int(plus_plus_int,J),L)) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_2501_add__le__less__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2502_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),D2)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2503_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2504_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_le_less_mono
tff(fact_2505_add__less__le__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2506_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),D2)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2507_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2508_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),D2)) ) ) ).

% add_less_le_mono
tff(fact_2509_add__neg__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_neg_neg
tff(fact_2510_add__neg__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_neg
tff(fact_2511_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_neg
tff(fact_2512_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% add_neg_neg
tff(fact_2513_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),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,A),B)),zero_zero_int) ) ) ).

% add_neg_neg
tff(fact_2514_add__pos__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_pos_pos
tff(fact_2515_add__pos__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_pos
tff(fact_2516_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_pos
tff(fact_2517_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% add_pos_pos
tff(fact_2518_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% add_pos_pos
tff(fact_2519_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ~ ! [C3: nat] :
            ( ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C3) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_2520_pos__add__strict,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% pos_add_strict
tff(fact_2521_pos__add__strict,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% pos_add_strict
tff(fact_2522_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(B),C)
       => aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2523_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% pos_add_strict
tff(fact_2524_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% pos_add_strict
tff(fact_2525_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),one_one_Code_integer)) ) ).

% add_mono1
tff(fact_2526_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),one_one_rat)) ) ).

% add_mono1
tff(fact_2527_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),one_one_nat)) ) ).

% add_mono1
tff(fact_2528_add__mono1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)) ) ).

% add_mono1
tff(fact_2529_less__add__one,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)) ).

% less_add_one
tff(fact_2530_less__add__one,axiom,
    ! [A: rat] : aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),one_one_rat)) ).

% less_add_one
tff(fact_2531_less__add__one,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)) ).

% less_add_one
tff(fact_2532_less__add__one,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)) ).

% less_add_one
tff(fact_2533_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
    <=> ( B = aa_int_int(uminus_uminus_int,A) ) ) ).

% add_eq_0_iff
tff(fact_2534_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
    <=> ( B = uminus_uminus_rat(A) ) ) ).

% add_eq_0_iff
tff(fact_2535_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
    <=> ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% add_eq_0_iff
tff(fact_2536_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),A) = zero_zero_int ).

% ab_group_add_class.ab_left_minus
tff(fact_2537_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),A) = zero_zero_rat ).

% ab_group_add_class.ab_left_minus
tff(fact_2538_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),A) = zero_z3403309356797280102nteger ).

% ab_group_add_class.ab_left_minus
tff(fact_2539_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int )
     => ( aa_int_int(uminus_uminus_int,A) = B ) ) ).

% add.inverse_unique
tff(fact_2540_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat )
     => ( uminus_uminus_rat(A) = B ) ) ).

% add.inverse_unique
tff(fact_2541_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger )
     => ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B ) ) ).

% add.inverse_unique
tff(fact_2542_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(uminus_uminus_int,B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2543_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = uminus_uminus_rat(B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2544_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2545_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(uminus_uminus_int,A) = B )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2546_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat(A) = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2547_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = B )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2548_sgn__minus__1,axiom,
    aa_int_int(sgn_sgn_int,aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% sgn_minus_1
tff(fact_2549_sgn__minus__1,axiom,
    sgn_sgn_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(one_one_rat) ).

% sgn_minus_1
tff(fact_2550_sgn__minus__1,axiom,
    aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% sgn_minus_1
tff(fact_2551_add_Ocomm__monoid__axioms,axiom,
    aa_nat_o(comm_monoid_nat(plus_plus_nat),zero_zero_nat) ).

% add.comm_monoid_axioms
tff(fact_2552_add_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(plus_plus_int),zero_zero_int) ).

% add.comm_monoid_axioms
tff(fact_2553_add_Ocomm__monoid__axioms,axiom,
    aa_rat_o(comm_monoid_rat(plus_plus_rat),zero_zero_rat) ).

% add.comm_monoid_axioms
tff(fact_2554_add_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).

% add.comm_monoid_axioms
tff(fact_2555_add_Ocomm__monoid__axioms,axiom,
    aa_Code_natural_o(comm_m478245049187884339atural(plus_p4538020629002901425atural),zero_z2226904508553997617atural) ).

% add.comm_monoid_axioms
tff(fact_2556_add_Omonoid__axioms,axiom,
    aa_nat_o(monoid_nat(plus_plus_nat),zero_zero_nat) ).

% add.monoid_axioms
tff(fact_2557_add_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(plus_plus_int),zero_zero_int) ).

% add.monoid_axioms
tff(fact_2558_add_Omonoid__axioms,axiom,
    aa_rat_o(monoid_rat(plus_plus_rat),zero_zero_rat) ).

% add.monoid_axioms
tff(fact_2559_add_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(plus_p5714425477246183910nteger),zero_z3403309356797280102nteger) ).

% add.monoid_axioms
tff(fact_2560_add_Omonoid__axioms,axiom,
    aa_Code_natural_o(monoid_Code_natural(plus_p4538020629002901425atural),zero_z2226904508553997617atural) ).

% add.monoid_axioms
tff(fact_2561_dbl__inc__def,axiom,
    ! [X: code_integer] : neg_nu5831290666863070958nteger(X) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X)),one_one_Code_integer) ).

% dbl_inc_def
tff(fact_2562_dbl__inc__def,axiom,
    ! [X: rat] : neg_nu5219082963157363817nc_rat(X) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_inc_def
tff(fact_2563_dbl__inc__def,axiom,
    ! [X: int] : aa_int_int(neg_nu5851722552734809277nc_int,X) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_inc_def
tff(fact_2564_add__neg__nonpos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_neg_nonpos
tff(fact_2565_add__neg__nonpos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_neg_nonpos
tff(fact_2566_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_neg_nonpos
tff(fact_2567_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% add_neg_nonpos
tff(fact_2568_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),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,A),B)),zero_zero_int) ) ) ).

% add_neg_nonpos
tff(fact_2569_add__nonneg__pos,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2570_add__nonneg__pos,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2571_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2572_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2573_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% add_nonneg_pos
tff(fact_2574_add__nonpos__neg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(A),zero_z2226904508553997617atural)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),zero_z2226904508553997617atural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),zero_z2226904508553997617atural) ) ) ).

% add_nonpos_neg
tff(fact_2575_add__nonpos__neg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),zero_z3403309356797280102nteger)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),zero_z3403309356797280102nteger) ) ) ).

% add_nonpos_neg
tff(fact_2576_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(B),zero_zero_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),zero_zero_rat) ) ) ).

% add_nonpos_neg
tff(fact_2577_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),zero_zero_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,A),B)),zero_zero_nat) ) ) ).

% add_nonpos_neg
tff(fact_2578_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),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,A),B)),zero_zero_int) ) ) ).

% add_nonpos_neg
tff(fact_2579_add__pos__nonneg,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),B)
       => aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2580_add__pos__nonneg,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2581_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
       => aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2582_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),B)
       => 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,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2583_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
       => 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,A),B)) ) ) ).

% add_pos_nonneg
tff(fact_2584_add__strict__increasing,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(B),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2585_add__strict__increasing,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2586_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(B),C)
       => aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2587_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2588_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing
tff(fact_2589_add__strict__increasing2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),C)
       => aa_Code_natural_o(ord_le5570908160329646204atural(B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2590_add__strict__increasing2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2591_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(B),C)
       => aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2592_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),C)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2593_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,B),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)) ) ) ).

% add_strict_increasing2
tff(fact_2594_sum__squares__ge__zero,axiom,
    ! [X: code_integer,Y: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2595_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2596_sum__squares__ge__zero,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))) ).

% sum_squares_ge_zero
tff(fact_2597_not__sum__squares__lt__zero,axiom,
    ! [X: code_integer,Y: code_integer] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger) ).

% not_sum_squares_lt_zero
tff(fact_2598_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] : ~ aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat) ).

% not_sum_squares_lt_zero
tff(fact_2599_not__sum__squares__lt__zero,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int) ).

% not_sum_squares_lt_zero
tff(fact_2600_zero__less__two,axiom,
    aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),one_one_Code_natural)) ).

% zero_less_two
tff(fact_2601_zero__less__two,axiom,
    aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer)) ).

% zero_less_two
tff(fact_2602_zero__less__two,axiom,
    aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat)) ).

% zero_less_two
tff(fact_2603_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_2604_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_2605_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = one_one_Code_integer )
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A) ) ).

% sgn_1_pos
tff(fact_2606_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = one_one_rat )
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).

% sgn_1_pos
tff(fact_2607_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = one_one_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A) ) ).

% sgn_1_pos
tff(fact_2608_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(X),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(Y),A)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),U)
         => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),V2)
           => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),V2) = one_one_Code_integer )
             => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,U),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2609_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( aa_rat_o(ord_less_eq_rat(X),A)
     => ( aa_rat_o(ord_less_eq_rat(Y),A)
       => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),U)
         => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),V2)
           => ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),V2) = one_one_rat )
             => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,U),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2610_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Y),A)
       => ( 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),V2)
           => ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,U),V2) = 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),X)),aa_int_int(aa_int_fun_int_int(times_times_int,V2),Y))),A) ) ) ) ) ) ).

% convex_bound_le
tff(fact_2611_sgn__if,axiom,
    ! [X: int] :
      aa_int_int(sgn_sgn_int,X) = $ite(
        X = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% sgn_if
tff(fact_2612_sgn__if,axiom,
    ! [X: rat] :
      sgn_sgn_rat(X) = $ite(
        X = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(ord_less_rat(zero_zero_rat),X),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_if
tff(fact_2613_sgn__if,axiom,
    ! [X: code_integer] :
      aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X) = $ite(
        X = zero_z3403309356797280102nteger,
        zero_z3403309356797280102nteger,
        $ite(aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),X),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ) ).

% sgn_if
tff(fact_2614_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( aa_int_int(sgn_sgn_int,A) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int) ) ).

% sgn_1_neg
tff(fact_2615_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat(A) = uminus_uminus_rat(one_one_rat) )
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% sgn_1_neg
tff(fact_2616_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger) ) ).

% sgn_1_neg
tff(fact_2617_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_2618_discrete,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer)),B) ) ).

% discrete
tff(fact_2619_discrete,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),B) ) ).

% discrete
tff(fact_2620_discrete,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),B) ) ).

% discrete
tff(fact_2621_sum__squares__gt__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)))
    <=> ( ( X != zero_z3403309356797280102nteger )
        | ( Y != zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2622_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)))
    <=> ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2623_sum__squares__gt__zero__iff,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)))
    <=> ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_2624_sum__squares__le__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y))),zero_z3403309356797280102nteger)
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2625_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y))),zero_zero_rat)
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2626_sum__squares__le__zero__iff,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y))),zero_zero_int)
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_2627_n__lists__Nil,axiom,
    ! [N: nat] :
      n_list679713369942834879n_assn(N,nil_Pr5671120429643327159n_assn) = $ite(N = zero_zero_nat,cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn),nil_li5476096274760905021n_assn) ).

% n_lists_Nil
tff(fact_2628_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_int(N,nil_int) = $ite(N = zero_zero_nat,cons_list_int(nil_int,nil_list_int),nil_list_int) ).

% n_lists_Nil
tff(fact_2629_n__lists__Nil,axiom,
    ! [N: nat] :
      n_lists_nat(N,nil_nat) = $ite(N = zero_zero_nat,cons_list_nat(nil_nat,nil_list_nat),nil_list_nat) ).

% n_lists_Nil
tff(fact_2630_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D2: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),C)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2631_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D2: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),C)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2632_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D2: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,R2),C)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2633_add__scale__eq__noteq,axiom,
    ! [R2: code_integer,A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( R2 != zero_z3403309356797280102nteger )
     => ( ( ( A = B )
          & ( C != D2 ) )
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),C)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,R2),D2)) ) ) ) ).

% add_scale_eq_noteq
tff(fact_2634_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = zero_zero_rat )
    <=> ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2635_sum__squares__eq__zero__iff,axiom,
    ! [X: 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,X),X)),aa_int_int(aa_int_fun_int_int(times_times_int,Y),Y)) = zero_zero_int )
    <=> ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2636_sum__squares__eq__zero__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = zero_z3403309356797280102nteger )
    <=> ( ( X = zero_z3403309356797280102nteger )
        & ( Y = zero_z3403309356797280102nteger ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_2637_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn] : n_list679713369942834879n_assn(zero_zero_nat,Xs) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% n_lists.simps(1)
tff(fact_2638_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_int] : n_lists_int(zero_zero_nat,Xs) = cons_list_int(nil_int,nil_list_int) ).

% n_lists.simps(1)
tff(fact_2639_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] : n_lists_nat(zero_zero_nat,Xs) = cons_list_nat(nil_nat,nil_list_nat) ).

% n_lists.simps(1)
tff(fact_2640_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),D2)) != aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),D2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2641_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2642_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),D2)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2643_crossproduct__noteq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( ( A != B )
        & ( C != D2 ) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% crossproduct_noteq
tff(fact_2644_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X: rat,Z: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2645_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Y)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,W),Z)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2646_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Y)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Z)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,W),Z)),aa_int_int(aa_int_fun_int_int(times_times_int,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2647_crossproduct__eq,axiom,
    ! [W: code_integer,Y: code_integer,X: code_integer,Z: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Y)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,W),Z)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y)) )
    <=> ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
tff(fact_2648_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_2649_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = one_one_nat )
    <=> ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_2650_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_2651_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_2652_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
tff(fact_2653_product__lists_Osimps_I1_J,axiom,
    produc1566369527784499744n_assn(nil_li5476096274760905021n_assn) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% product_lists.simps(1)
tff(fact_2654_product__lists_Osimps_I1_J,axiom,
    product_lists_int(nil_list_int) = cons_list_int(nil_int,nil_list_int) ).

% product_lists.simps(1)
tff(fact_2655_product__lists_Osimps_I1_J,axiom,
    product_lists_nat(nil_list_nat) = cons_list_nat(nil_nat,nil_list_nat) ).

% product_lists.simps(1)
tff(fact_2656_power__decreasing__iff,axiom,
    ! [B: code_natural,M2: nat,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),one_one_Code_natural)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(power_7079662738309270450atural(B,M2)),power_7079662738309270450atural(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_2657_power__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),one_one_Code_integer)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(B,M2)),power_8256067586552552935nteger(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_2658_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
     => ( aa_rat_o(ord_less_rat(B),one_one_rat)
       => ( aa_rat_o(ord_less_eq_rat(power_power_rat(B,M2)),power_power_rat(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_2659_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_2660_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ) ) ) ).

% power_decreasing_iff
tff(fact_2661_subseqs_Osimps_I1_J,axiom,
    subseq184808802919281286n_assn(nil_Pr5671120429643327159n_assn) = cons_l2423627976422276333n_assn(nil_Pr5671120429643327159n_assn,nil_li5476096274760905021n_assn) ).

% subseqs.simps(1)
tff(fact_2662_subseqs_Osimps_I1_J,axiom,
    subseqs_int(nil_int) = cons_list_int(nil_int,nil_list_int) ).

% subseqs.simps(1)
tff(fact_2663_subseqs_Osimps_I1_J,axiom,
    subseqs_nat(nil_nat) = cons_list_nat(nil_nat,nil_list_nat) ).

% subseqs.simps(1)
tff(fact_2664_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)) ) ) ).

% le_minus_divide_eq
tff(fact_2665_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)) ) ) ).

% minus_divide_le_eq
tff(fact_2666_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2667_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),C) ).

% times_divide_eq_right
tff(fact_2668_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ).

% divide_divide_eq_right
tff(fact_2669_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ).

% divide_divide_eq_left
tff(fact_2670_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),A)),C) ).

% times_divide_eq_left
tff(fact_2671_div__by__1,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,one_one_Code_integer) = A ).

% div_by_1
tff(fact_2672_div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% div_by_1
tff(fact_2673_div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% div_by_1
tff(fact_2674_div__by__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),one_one_rat) = A ).

% div_by_1
tff(fact_2675_div__by__1,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,one_one_Code_natural) = A ).

% div_by_1
tff(fact_2676_power__one,axiom,
    ! [N: nat] : power_8256067586552552935nteger(one_one_Code_integer,N) = one_one_Code_integer ).

% power_one
tff(fact_2677_power__one,axiom,
    ! [N: nat] : aa_nat_assn(power_power_assn(one_one_assn),N) = one_one_assn ).

% power_one
tff(fact_2678_power__one,axiom,
    ! [N: nat] : power_power_rat(one_one_rat,N) = one_one_rat ).

% power_one
tff(fact_2679_power__one,axiom,
    ! [N: nat] : aa_nat_int(power_power_int(one_one_int),N) = one_one_int ).

% power_one
tff(fact_2680_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_2681_power__one__right,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),one_one_nat) = A ).

% power_one_right
tff(fact_2682_power__one__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),one_one_nat) = A ).

% power_one_right
tff(fact_2683_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_2684_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_2685_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_2686_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_2687_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) = $ite(C = zero_zero_rat,zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) ).

% mult_divide_mult_cancel_left_if
tff(fact_2688_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2689_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2690_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != 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,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2691_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2692_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),B) = A ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_2693_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2694_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != 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,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2695_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != 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,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2696_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2697_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),A) = B ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_2698_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) )
    <=> ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
tff(fact_2699_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
tff(fact_2700_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
tff(fact_2701_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A) = one_one_rat )
    <=> ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
tff(fact_2702_divide__self__if,axiom,
    ! [A: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% divide_self_if
tff(fact_2703_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% divide_self
tff(fact_2704_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
tff(fact_2705_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
    <=> ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
tff(fact_2706_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(A,A) = one_one_Code_integer ) ) ).

% div_self
tff(fact_2707_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),A) = one_one_nat ) ) ).

% div_self
tff(fact_2708_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),A) = one_one_int ) ) ).

% div_self
tff(fact_2709_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),A) = one_one_rat ) ) ).

% div_self
tff(fact_2710_div__self,axiom,
    ! [A: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(A,A) = one_one_Code_natural ) ) ).

% div_self
tff(fact_2711_power__inject__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => ( ( power_8256067586552552935nteger(A,M2) = power_8256067586552552935nteger(A,N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2712_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => ( ( power_power_rat(A,M2) = power_power_rat(A,N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2713_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2714_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( ( aa_nat_int(power_power_int(A),M2) = aa_nat_int(power_power_int(A),N) )
      <=> ( M2 = N ) ) ) ).

% power_inject_exp
tff(fact_2715_divide__minus1,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(X) ).

% divide_minus1
tff(fact_2716_divide__sgn,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),sgn_sgn_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),sgn_sgn_rat(B)) ).

% divide_sgn
tff(fact_2717_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).

% zero_le_divide_1_iff
tff(fact_2718_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) ) ).

% divide_le_0_1_iff
tff(fact_2719_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A))
    <=> aa_rat_o(ord_less_rat(zero_zero_rat),A) ) ).

% zero_less_divide_1_iff
tff(fact_2720_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(ord_less_rat(A),B) ) ) ).

% less_divide_eq_1_pos
tff(fact_2721_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(ord_less_rat(B),A) ) ) ).

% less_divide_eq_1_neg
tff(fact_2722_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(ord_less_rat(B),A) ) ) ).

% divide_less_eq_1_pos
tff(fact_2723_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(ord_less_rat(A),B) ) ) ).

% divide_less_eq_1_neg
tff(fact_2724_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A)),zero_zero_rat)
    <=> aa_rat_o(ord_less_rat(A),zero_zero_rat) ) ).

% divide_less_0_1_iff
tff(fact_2725_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_2726_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),B) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_2727_power__strict__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(B,X)),power_8256067586552552935nteger(B,Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2728_power__strict__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),B)
     => ( aa_rat_o(ord_less_rat(power_power_rat(B,X)),power_power_rat(B,Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2729_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2730_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Y) ) ) ).

% power_strict_increasing_iff
tff(fact_2731_left__minus__one__mult__self,axiom,
    ! [N: nat,A: 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)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2732_left__minus__one__mult__self,axiom,
    ! [N: nat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(one_one_rat),N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(one_one_rat),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2733_left__minus__one__mult__self,axiom,
    ! [N: nat,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)),A)) = A ).

% left_minus_one_mult_self
tff(fact_2734_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_2735_minus__one__mult__self,axiom,
    ! [N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(one_one_rat),N)),power_power_rat(uminus_uminus_rat(one_one_rat),N)) = one_one_rat ).

% minus_one_mult_self
tff(fact_2736_minus__one__mult__self,axiom,
    ! [N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)) = one_one_Code_integer ).

% minus_one_mult_self
tff(fact_2737_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).

% le_divide_eq_1_pos
tff(fact_2738_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
      <=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).

% le_divide_eq_1_neg
tff(fact_2739_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(ord_less_eq_rat(B),A) ) ) ).

% divide_le_eq_1_pos
tff(fact_2740_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
      <=> aa_rat_o(ord_less_eq_rat(A),B) ) ) ).

% divide_le_eq_1_neg
tff(fact_2741_power__strict__decreasing__iff,axiom,
    ! [B: code_natural,M2: nat,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),B)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(B),one_one_Code_natural)
       => ( aa_Code_natural_o(ord_le5570908160329646204atural(power_7079662738309270450atural(B,M2)),power_7079662738309270450atural(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2742_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),one_one_Code_integer)
       => ( aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(B,M2)),power_8256067586552552935nteger(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2743_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),B)
     => ( aa_rat_o(ord_less_rat(B),one_one_rat)
       => ( aa_rat_o(ord_less_rat(power_power_rat(B,M2)),power_power_rat(B,N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2744_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),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,B),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2745_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),one_one_int)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(B),M2)),aa_nat_int(power_power_int(B),N))
        <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_2746_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(B,X)),power_8256067586552552935nteger(B,Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2747_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),B)
     => ( aa_rat_o(ord_less_eq_rat(power_power_rat(B,X)),power_power_rat(B,Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2748_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),X)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2749_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(B),X)),aa_nat_int(power_power_int(B),Y))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,X),Y) ) ) ).

% power_increasing_iff
tff(fact_2750_power__one__over,axiom,
    ! [A: rat,N: nat] : power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),A),N) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),power_power_rat(A,N)) ).

% power_one_over
tff(fact_2751_power__add,axiom,
    ! [A: assn,M2: nat,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),M2)),aa_nat_assn(power_power_assn(A),N)) ).

% power_add
tff(fact_2752_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_add
tff(fact_2753_power__add,axiom,
    ! [A: int,M2: nat,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) ).

% power_add
tff(fact_2754_power__add,axiom,
    ! [A: code_integer,M2: nat,N: nat] : power_8256067586552552935nteger(A,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,M2)),power_8256067586552552935nteger(A,N)) ).

% power_add
tff(fact_2755_power__add,axiom,
    ! [A: code_natural,M2: nat,N: nat] : power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,M2)),power_7079662738309270450atural(A,N)) ).

% power_add
tff(fact_2756_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),X) )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,Y),aa_nat_assn(power_power_assn(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2757_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),Y) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Y),X) )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),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,X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2758_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),X) )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),Y) = aa_int_int(aa_int_fun_int_int(times_times_int,Y),aa_nat_int(power_power_int(X),N)) ) ) ).

% power_commuting_commutes
tff(fact_2759_power__commuting__commutes,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X) )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(X,N)),Y) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),power_8256067586552552935nteger(X,N)) ) ) ).

% power_commuting_commutes
tff(fact_2760_power__commuting__commutes,axiom,
    ! [X: code_natural,Y: code_natural,N: nat] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Y) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Y),X) )
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(X,N)),Y) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Y),power_7079662738309270450atural(X,N)) ) ) ).

% power_commuting_commutes
tff(fact_2761_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] : aa_nat_assn(power_power_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),N) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),aa_nat_assn(power_power_assn(B),N)) ).

% power_mult_distrib
tff(fact_2762_power__mult__distrib,axiom,
    ! [A: nat,B: 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,A),B)),N) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,B),N)) ).

% power_mult_distrib
tff(fact_2763_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),N) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(B),N)) ).

% power_mult_distrib
tff(fact_2764_power__mult__distrib,axiom,
    ! [A: code_integer,B: code_integer,N: nat] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,N)),power_8256067586552552935nteger(B,N)) ).

% power_mult_distrib
tff(fact_2765_power__mult__distrib,axiom,
    ! [A: code_natural,B: code_natural,N: nat] : power_7079662738309270450atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),N) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,N)),power_7079662738309270450atural(B,N)) ).

% power_mult_distrib
tff(fact_2766_power__commutes,axiom,
    ! [A: assn,N: nat] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).

% power_commutes
tff(fact_2767_power__commutes,axiom,
    ! [A: 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,A),N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_commutes
tff(fact_2768_power__commutes,axiom,
    ! [A: int,N: nat] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_commutes
tff(fact_2769_power__commutes,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,N)),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(A,N)) ).

% power_commutes
tff(fact_2770_power__commutes,axiom,
    ! [A: code_natural,N: nat] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,N)),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N)) ).

% power_commutes
tff(fact_2771_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B)) ).

% divide_divide_eq_left'
tff(fact_2772_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),W)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ).

% divide_divide_times_eq
tff(fact_2773_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z: rat,W: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Z),W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),W)) ).

% times_divide_times_eq
tff(fact_2774_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),A)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),power_8256067586552552935nteger(A,N)) ) ).

% one_le_power
tff(fact_2775_one__le__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
     => aa_rat_o(ord_less_eq_rat(one_one_rat),power_power_rat(A,N)) ) ).

% one_le_power
tff(fact_2776_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => 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,A),N)) ) ).

% one_le_power
tff(fact_2777_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_nat_int(power_power_int(A),N)) ) ).

% one_le_power
tff(fact_2778_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Y) = one_one_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(X,N)),power_power_rat(Y,N)) = one_one_rat ) ) ).

% left_right_inverse_power
tff(fact_2779_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,X),Y) = one_one_assn )
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(X),N)),aa_nat_assn(power_power_assn(Y),N)) = one_one_assn ) ) ).

% left_right_inverse_power
tff(fact_2780_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,X),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,X),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),N)) = one_one_nat ) ) ).

% left_right_inverse_power
tff(fact_2781_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,X),Y) = one_one_int )
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(X),N)),aa_nat_int(power_power_int(Y),N)) = one_one_int ) ) ).

% left_right_inverse_power
tff(fact_2782_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),Y) = one_one_Code_integer )
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(X,N)),power_8256067586552552935nteger(Y,N)) = one_one_Code_integer ) ) ).

% left_right_inverse_power
tff(fact_2783_left__right__inverse__power,axiom,
    ! [X: code_natural,Y: code_natural,N: nat] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),Y) = one_one_Code_natural )
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(X,N)),power_7079662738309270450atural(Y,N)) = one_one_Code_natural ) ) ).

% left_right_inverse_power
tff(fact_2784_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_2785_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
      <=> ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_2786_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B )
       => ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) ) ) ) ).

% eq_divide_imp
tff(fact_2787_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A ) ) ) ).

% divide_eq_imp
tff(fact_2788_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = B,A = zero_zero_rat) ) ).

% eq_divide_eq
tff(fact_2789_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = A )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% divide_eq_eq
tff(fact_2790_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z) )
        <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y) ) ) ) ) ).

% frac_eq_eq
tff(fact_2791_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = one_one_rat )
      <=> ( A = B ) ) ) ).

% right_inverse_eq
tff(fact_2792_power__0,axiom,
    ! [A: code_integer] : power_8256067586552552935nteger(A,zero_zero_nat) = one_one_Code_integer ).

% power_0
tff(fact_2793_power__0,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),zero_zero_nat) = one_one_assn ).

% power_0
tff(fact_2794_power__0,axiom,
    ! [A: rat] : power_power_rat(A,zero_zero_nat) = one_one_rat ).

% power_0
tff(fact_2795_power__0,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),zero_zero_nat) = one_one_int ).

% power_0
tff(fact_2796_power__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),zero_zero_nat) = one_one_nat ).

% power_0
tff(fact_2797_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(A,N)),one_one_Code_integer) ) ) ).

% power_le_one
tff(fact_2798_power__le__one,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(A),one_one_Code_natural)
       => aa_Code_natural_o(ord_le1926595141338095240atural(power_7079662738309270450atural(A,N)),one_one_Code_natural) ) ) ).

% power_le_one
tff(fact_2799_power__le__one,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
       => aa_rat_o(ord_less_eq_rat(power_power_rat(A,N)),one_one_rat) ) ) ).

% power_le_one
tff(fact_2800_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),N)),one_one_nat) ) ) ).

% power_le_one
tff(fact_2801_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),one_one_int) ) ) ).

% power_le_one
tff(fact_2802_power__less__power__Suc,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),A)
     => aa_Code_natural_o(ord_le5570908160329646204atural(power_7079662738309270450atural(A,N)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N))) ) ).

% power_less_power_Suc
tff(fact_2803_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(A,N)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(A,N))) ) ).

% power_less_power_Suc
tff(fact_2804_power__less__power__Suc,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => aa_rat_o(ord_less_rat(power_power_rat(A,N)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),power_power_rat(A,N))) ) ).

% power_less_power_Suc
tff(fact_2805_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_less_power_Suc
tff(fact_2806_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N))) ) ).

% power_less_power_Suc
tff(fact_2807_power__gt1__lemma,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),A)
     => aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N))) ) ).

% power_gt1_lemma
tff(fact_2808_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(A,N))) ) ).

% power_gt1_lemma
tff(fact_2809_power__gt1__lemma,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),power_power_rat(A,N))) ) ).

% power_gt1_lemma
tff(fact_2810_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => 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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))) ) ).

% power_gt1_lemma
tff(fact_2811_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => 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,A),aa_nat_int(power_power_int(A),N))) ) ).

% power_gt1_lemma
tff(fact_2812_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2813_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2814_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_2815_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_2816_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_strict_left_mono
tff(fact_2817_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
       => aa_rat_o(ord_less_rat(Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_less_div_pos
tff(fact_2818_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
     => ( aa_rat_o(ord_less_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) ) ) ).

% mult_imp_div_pos_less
tff(fact_2819_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_less_divide_eq
tff(fact_2820_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_less_eq
tff(fact_2821_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_divide_eq
tff(fact_2822_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_less_eq
tff(fact_2823_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_rat(A),zero_zero_rat)) ) ) ).

% less_divide_eq
tff(fact_2824_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),A)) ) ) ).

% divide_less_eq
tff(fact_2825_power__0__left,axiom,
    ! [N: nat] :
      power_power_rat(zero_zero_rat,N) = $ite(N = zero_zero_nat,one_one_rat,zero_zero_rat) ).

% power_0_left
tff(fact_2826_power__0__left,axiom,
    ! [N: nat] :
      power_8256067586552552935nteger(zero_z3403309356797280102nteger,N) = $ite(N = zero_zero_nat,one_one_Code_integer,zero_z3403309356797280102nteger) ).

% power_0_left
tff(fact_2827_power__0__left,axiom,
    ! [N: nat] :
      power_7079662738309270450atural(zero_z2226904508553997617atural,N) = $ite(N = zero_zero_nat,one_one_Code_natural,zero_z2226904508553997617atural) ).

% power_0_left
tff(fact_2828_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_2829_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_2830_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_rat(A),B) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_rat(B),A) ) ) ) ).

% less_divide_eq_1
tff(fact_2831_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_rat(B),A) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_rat(A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
tff(fact_2832_divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_add_eq_iff
tff(fact_2833_add__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% add_divide_eq_iff
tff(fact_2834_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_num_frac
tff(fact_2835_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))),Y) ) ) ).

% add_frac_num
tff(fact_2836_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% add_frac_eq
tff(fact_2837_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(1)
tff(fact_2838_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(2)
tff(fact_2839_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(A,N)),power_8256067586552552935nteger(A,N2)) ) ) ).

% power_strict_increasing
tff(fact_2840_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_rat_o(ord_less_rat(one_one_rat),A)
       => aa_rat_o(ord_less_rat(power_power_rat(A,N)),power_power_rat(A,N2)) ) ) ).

% power_strict_increasing
tff(fact_2841_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_strict_increasing
tff(fact_2842_power__strict__increasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_strict_increasing
tff(fact_2843_power__less__imp__less__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(A,M2)),power_8256067586552552935nteger(A,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_2844_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => ( aa_rat_o(ord_less_rat(power_power_rat(A,M2)),power_power_rat(A,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_2845_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_2846_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N) ) ) ).

% power_less_imp_less_exp
tff(fact_2847_power__minus,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),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(A),N)) ).

% power_minus
tff(fact_2848_power__minus,axiom,
    ! [A: rat,N: nat] : power_power_rat(uminus_uminus_rat(A),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(one_one_rat),N)),power_power_rat(A,N)) ).

% power_minus
tff(fact_2849_power__minus,axiom,
    ! [A: code_integer,N: nat] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)),power_8256067586552552935nteger(A,N)) ).

% power_minus
tff(fact_2850_power__increasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),A)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(A,N)),power_8256067586552552935nteger(A,N2)) ) ) ).

% power_increasing
tff(fact_2851_power__increasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
       => aa_rat_o(ord_less_eq_rat(power_power_rat(A,N)),power_power_rat(A,N2)) ) ) ).

% power_increasing
tff(fact_2852_power__increasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N2)) ) ) ).

% power_increasing
tff(fact_2853_power__increasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N)),aa_nat_int(power_power_int(A),N2)) ) ) ).

% power_increasing
tff(fact_2854_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))) ) ).

% less_half_sum
tff(fact_2855_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat))),B) ) ).

% gt_half_sum
tff(fact_2856_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) )
      <=> ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) = uminus_uminus_rat(A) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2857_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B)) = C )
      <=> ( uminus_uminus_rat(A) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2858_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) = A )
    <=> $ite(C != zero_zero_rat,uminus_uminus_rat(B) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C),A = zero_zero_rat) ) ).

% minus_divide_eq_eq
tff(fact_2859_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C) = uminus_uminus_rat(B),A = zero_zero_rat) ) ).

% eq_minus_divide_eq
tff(fact_2860_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),B) = uminus_uminus_rat(one_one_rat) )
    <=> ( ( B != zero_zero_rat )
        & ( A = uminus_uminus_rat(B) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2861_power__Suc__less,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(A),one_one_Code_natural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N))),power_7079662738309270450atural(A,N)) ) ) ).

% power_Suc_less
tff(fact_2862_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),one_one_Code_integer)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(A,N))),power_8256067586552552935nteger(A,N)) ) ) ).

% power_Suc_less
tff(fact_2863_power__Suc__less,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(A),one_one_rat)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),power_power_rat(A,N))),power_power_rat(A,N)) ) ) ).

% power_Suc_less
tff(fact_2864_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% power_Suc_less
tff(fact_2865_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),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,A),aa_nat_int(power_power_int(A),N))),aa_nat_int(power_power_int(A),N)) ) ) ).

% power_Suc_less
tff(fact_2866_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(C),zero_zero_rat)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono_neg
tff(fact_2867_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y)),X)
       => aa_rat_o(ord_less_eq_rat(Z),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)) ) ) ).

% mult_imp_le_div_pos
tff(fact_2868_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),Y)
     => ( aa_rat_o(ord_less_eq_rat(X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),Y))
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),Z) ) ) ).

% mult_imp_div_pos_le
tff(fact_2869_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% pos_le_divide_eq
tff(fact_2870_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_divide_le_eq
tff(fact_2871_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
      <=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_le_divide_eq
tff(fact_2872_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
      <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B) ) ) ).

% neg_divide_le_eq
tff(fact_2873_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),C)
       => ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B))
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,C),B)) ) ) ) ).

% divide_left_mono
tff(fact_2874_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)) ) ) ).

% le_divide_eq
tff(fact_2875_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),A)
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)) ) ) ).

% divide_le_eq
tff(fact_2876_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A))
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(A),B) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(B),A) ) ) ) ).

% le_divide_eq_1
tff(fact_2877_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),A)),one_one_rat)
    <=> ( ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
          & aa_rat_o(ord_less_eq_rat(B),A) )
        | ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
          & aa_rat_o(ord_less_eq_rat(A),B) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
tff(fact_2878_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(ord_le5570908160329646204atural(A),one_one_Code_natural)
         => aa_Code_natural_o(ord_le5570908160329646204atural(power_7079662738309270450atural(A,N2)),power_7079662738309270450atural(A,N)) ) ) ) ).

% power_strict_decreasing
tff(fact_2879_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),one_one_Code_integer)
         => aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(A,N2)),power_8256067586552552935nteger(A,N)) ) ) ) ).

% power_strict_decreasing
tff(fact_2880_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),N2)
     => ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
       => ( aa_rat_o(ord_less_rat(A),one_one_rat)
         => aa_rat_o(ord_less_rat(power_power_rat(A,N2)),power_power_rat(A,N)) ) ) ) ).

% power_strict_decreasing
tff(fact_2881_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),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,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_2882_power__strict__decreasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_strict_decreasing
tff(fact_2883_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer)
         => aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(A,N2)),power_8256067586552552935nteger(A,N)) ) ) ) ).

% power_decreasing
tff(fact_2884_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
       => ( aa_Code_natural_o(ord_le1926595141338095240atural(A),one_one_Code_natural)
         => aa_Code_natural_o(ord_le1926595141338095240atural(power_7079662738309270450atural(A,N2)),power_7079662738309270450atural(A,N)) ) ) ) ).

% power_decreasing
tff(fact_2885_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: rat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),N2)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
       => ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
         => aa_rat_o(ord_less_eq_rat(power_power_rat(A,N2)),power_power_rat(A,N)) ) ) ) ).

% power_decreasing
tff(fact_2886_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),N2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ) ).

% power_decreasing
tff(fact_2887_power__decreasing,axiom,
    ! [N: nat,N2: nat,A: 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),A)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
         => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),N2)),aa_nat_int(power_power_int(A),N)) ) ) ) ).

% power_decreasing
tff(fact_2888_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(A,M2)),power_8256067586552552935nteger(A,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_2889_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(power_power_rat(A,M2)),power_power_rat(A,N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_2890_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_2891_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_le_imp_le_exp
tff(fact_2892_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(A),power_8256067586552552935nteger(A,N)) ) ) ).

% self_le_power
tff(fact_2893_self__le__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_eq_rat(one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(ord_less_eq_rat(A),power_power_rat(A,N)) ) ) ).

% self_le_power
tff(fact_2894_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),A)
     => ( 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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ) ) ).

% self_le_power
tff(fact_2895_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),A)
     => ( 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,A),aa_nat_int(power_power_int(A),N)) ) ) ).

% self_le_power
tff(fact_2896_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2897_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2898_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2899_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2900_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),aa_rat_o(ord_less_rat(zero_zero_rat),A)) ) ) ).

% minus_divide_less_eq
tff(fact_2901_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_o(ord_less_rat(A),zero_zero_rat)) ) ) ).

% less_minus_divide_eq
tff(fact_2902_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),power_8256067586552552935nteger(A,N)) ) ) ).

% one_less_power
tff(fact_2903_one__less__power,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => aa_rat_o(ord_less_rat(one_one_rat),power_power_rat(A,N)) ) ) ).

% one_less_power
tff(fact_2904_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => ( 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,A),N)) ) ) ).

% one_less_power
tff(fact_2905_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => ( 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(A),N)) ) ) ).

% one_less_power
tff(fact_2906_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2907_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,B,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(3)
tff(fact_2908_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2909_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),C)
     => ( aa_rat_o(ord_less_eq_rat(A),uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)))
      <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2910_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),zero_zero_rat)
     => ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))),A)
      <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),uminus_uminus_rat(B)) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2911_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self4
tff(fact_2912_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != 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,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self4
tff(fact_2913_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != 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,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self4
tff(fact_2914_div__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self4
tff(fact_2915_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self3
tff(fact_2916_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != 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),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self3
tff(fact_2917_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != 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),B)),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self3
tff(fact_2918_div__mult__self3,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self3
tff(fact_2919_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self2
tff(fact_2920_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self2
tff(fact_2921_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self2
tff(fact_2922_div__mult__self2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self2
tff(fact_2923_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),divide6298287555418463151nteger(A,B)) ) ) ).

% div_mult_self1
tff(fact_2924_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ) ) ).

% div_mult_self1
tff(fact_2925_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ) ) ).

% div_mult_self1
tff(fact_2926_div__mult__self1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),divide5121882707175180666atural(A,B)) ) ) ).

% div_mult_self1
tff(fact_2927_div__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,A) ).

% div_minus1_right
tff(fact_2928_div__minus1__right,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% div_minus1_right
tff(fact_2929_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = $ite(C = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,divide6298287555418463151nteger(A,B)) ).

% div_mult_mult1_if
tff(fact_2930_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = $ite(C = zero_zero_nat,zero_zero_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% div_mult_mult1_if
tff(fact_2931_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = $ite(C = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% div_mult_mult1_if
tff(fact_2932_div__mult__mult1__if,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) = $ite(C = zero_z2226904508553997617atural,zero_z2226904508553997617atural,divide5121882707175180666atural(A,B)) ).

% div_mult_mult1_if
tff(fact_2933_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult2
tff(fact_2934_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult2
tff(fact_2935_div__mult__mult2,axiom,
    ! [C: int,A: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult2
tff(fact_2936_div__mult__mult2,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult2
tff(fact_2937_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) = divide6298287555418463151nteger(A,B) ) ) ).

% div_mult_mult1
tff(fact_2938_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ).

% div_mult_mult1
tff(fact_2939_div__mult__mult1,axiom,
    ! [C: int,A: int,B: 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),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ).

% div_mult_mult1
tff(fact_2940_div__mult__mult1,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) = divide5121882707175180666atural(A,B) ) ) ).

% div_mult_mult1
tff(fact_2941_bits__div__by__1,axiom,
    ! [A: code_integer] : divide6298287555418463151nteger(A,one_one_Code_integer) = A ).

% bits_div_by_1
tff(fact_2942_bits__div__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),one_one_nat) = A ).

% bits_div_by_1
tff(fact_2943_bits__div__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(divide_divide_int,A),one_one_int) = A ).

% bits_div_by_1
tff(fact_2944_bits__div__by__1,axiom,
    ! [A: code_natural] : divide5121882707175180666atural(A,one_one_Code_natural) = A ).

% bits_div_by_1
tff(fact_2945_div__less__dividend,axiom,
    ! [N: nat,M2: 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),M2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N)),M2) ) ) ).

% div_less_dividend
tff(fact_2946_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,M2),N) = M2 )
      <=> ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
tff(fact_2947_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% div_add_self1
tff(fact_2948_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,B),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self1
tff(fact_2949_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != 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,B),A)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self1
tff(fact_2950_div__add__self1,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% div_add_self1
tff(fact_2951_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% div_add_self2
tff(fact_2952_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != 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,A),B)),B) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),one_one_nat) ) ) ).

% div_add_self2
tff(fact_2953_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != 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,A),B)),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int) ) ) ).

% div_add_self2
tff(fact_2954_div__add__self2,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% div_add_self2
tff(fact_2955_power__minus_H,axiom,
    ! [X: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,one_one_int),X)
     => ( aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,X)),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(X),N)) ) ) ).

% power_minus'
tff(fact_2956_power__minus_H,axiom,
    ! [X: rat,N: nat] :
      ( aa_rat_o(nO_MATCH_rat_rat(one_one_rat),X)
     => ( power_power_rat(uminus_uminus_rat(X),N) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(one_one_rat),N)),power_power_rat(X,N)) ) ) ).

% power_minus'
tff(fact_2957_power__minus_H,axiom,
    ! [X: code_integer,N: nat] :
      ( aa_Code_integer_o(nO_MAT8252062027627875367nteger(one_one_Code_integer),X)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X),N) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N)),power_8256067586552552935nteger(X,N)) ) ) ).

% power_minus'
tff(fact_2958_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2959_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(numeral_numeral_rat(W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2960_count__list_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      aa_int_nat(count_list_int(cons_int(X,Xs)),Y) = $ite(X = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_int_nat(count_list_int(Xs),Y)),one_one_nat),aa_int_nat(count_list_int(Xs),Y)) ).

% count_list.simps(2)
tff(fact_2961_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      aa_nat_nat(count_list_nat(cons_nat(X,Xs)),Y) = $ite(X = Y,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(count_list_nat(Xs),Y)),one_one_nat),aa_nat_nat(count_list_nat(Xs),Y)) ).

% count_list.simps(2)
tff(fact_2962_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2963_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2964_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2965_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2966_distrib__right__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(nO_MAT7518554365073749349atural(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2967_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: rat,A: rat,B: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2968_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2969_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2970_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2971_distrib__right__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(nO_MAT7890207925310593985atural(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ) ).

% distrib_right_NO_MATCH
tff(fact_2972_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_nat_rat(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2973_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(nO_MATCH_nat_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2974_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2975_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2976_distrib__left__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(nO_MAT7518554365073749349atural(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2977_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: rat,B: rat,C: rat] :
      ( aa_rat_o(nO_MATCH_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2978_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(nO_MATCH_int_nat,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2979_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2980_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2981_distrib__left__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(nO_MAT7890207925310593985atural(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)) ) ) ).

% distrib_left_NO_MATCH
tff(fact_2982_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_assn(power_power_assn(A),N) ) ) ).

% power_minus_mult
tff(fact_2983_power__minus__mult,axiom,
    ! [N: nat,A: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N) ) ) ).

% power_minus_mult
tff(fact_2984_power__minus__mult,axiom,
    ! [N: nat,A: 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(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = aa_nat_int(power_power_int(A),N) ) ) ).

% power_minus_mult
tff(fact_2985_power__minus__mult,axiom,
    ! [N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = power_8256067586552552935nteger(A,N) ) ) ).

% power_minus_mult
tff(fact_2986_power__minus__mult,axiom,
    ! [N: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))),A) = power_7079662738309270450atural(A,N) ) ) ).

% power_minus_mult
tff(fact_2987_diff__self,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% diff_self
tff(fact_2988_diff__self,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% diff_self
tff(fact_2989_diff__self,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,A) = zero_z3403309356797280102nteger ).

% diff_self
tff(fact_2990_diff__0__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_0_right
tff(fact_2991_diff__0__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_0_right
tff(fact_2992_diff__0__right,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,zero_z3403309356797280102nteger) = A ).

% diff_0_right
tff(fact_2993_zero__diff,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,zero_zero_nat),A) = zero_zero_nat ).

% zero_diff
tff(fact_2994_zero__diff,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(zero_z2226904508553997617atural,A) = zero_z2226904508553997617atural ).

% zero_diff
tff(fact_2995_diff__zero,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),zero_zero_rat) = A ).

% diff_zero
tff(fact_2996_diff__zero,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),zero_zero_nat) = A ).

% diff_zero
tff(fact_2997_diff__zero,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),zero_zero_int) = A ).

% diff_zero
tff(fact_2998_diff__zero,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,zero_z3403309356797280102nteger) = A ).

% diff_zero
tff(fact_2999_diff__zero,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(A,zero_z2226904508553997617atural) = A ).

% diff_zero
tff(fact_3000_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),A) = zero_zero_rat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3001_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),A) = zero_zero_nat ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3002_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),A) = zero_zero_int ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3003_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(A,A) = zero_z3403309356797280102nteger ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3004_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: code_natural] : minus_7197305767214868737atural(A,A) = zero_z2226904508553997617atural ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_3005_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V2)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(W)),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(times_times_num(V2,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3006_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),Z)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(times_times_num(V2,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3007_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V2,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3008_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Z)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(times_times_num(V2,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3009_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V2)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(W)),Z)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(times_times_num(V2,W))),Z) ).

% mult_numeral_left_semiring_numeral
tff(fact_3010_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)) = numeral_numeral_nat(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3011_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(M2)),numeral_numeral_int(N)) = numeral_numeral_int(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3012_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M2)),numeral_numeral_rat(N)) = numeral_numeral_rat(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3013_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M2)),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3014_numeral__times__numeral,axiom,
    ! [M2: num,N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(M2)),numera5444537566228673987atural(N)) = numera5444537566228673987atural(times_times_num(M2,N)) ).

% numeral_times_numeral
tff(fact_3015_add__diff__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3016_add__diff__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel
tff(fact_3017_add__diff__cancel,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),B) = A ).

% add_diff_cancel
tff(fact_3018_diff__add__cancel,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3019_diff__add__cancel,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),B) = A ).

% diff_add_cancel
tff(fact_3020_diff__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,minus_8373710615458151222nteger(A,B)),B) = A ).

% diff_add_cancel
tff(fact_3021_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_left
tff(fact_3022_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_left
tff(fact_3023_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,C),A)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_left
tff(fact_3024_add__diff__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) = minus_8373710615458151222nteger(A,B) ).

% add_diff_cancel_left
tff(fact_3025_add__diff__cancel__left,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] : minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),B)) = minus_7197305767214868737atural(A,B) ).

% add_diff_cancel_left
tff(fact_3026_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3027_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3028_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),A) = B ).

% add_diff_cancel_left'
tff(fact_3029_add__diff__cancel__left_H,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_3030_add__diff__cancel__left_H,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),A) = B ).

% add_diff_cancel_left'
tff(fact_3031_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ).

% add_diff_cancel_right
tff(fact_3032_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ).

% add_diff_cancel_right
tff(fact_3033_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ).

% add_diff_cancel_right
tff(fact_3034_add__diff__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = minus_8373710615458151222nteger(A,B) ).

% add_diff_cancel_right
tff(fact_3035_add__diff__cancel__right,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = minus_7197305767214868737atural(A,B) ).

% add_diff_cancel_right
tff(fact_3036_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3037_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3038_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),B) = A ).

% add_diff_cancel_right'
tff(fact_3039_add__diff__cancel__right_H,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_3040_add__diff__cancel__right_H,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B),B) = A ).

% add_diff_cancel_right'
tff(fact_3041_minus__diff__eq,axiom,
    ! [A: int,B: int] : aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% minus_diff_eq
tff(fact_3042_minus__diff__eq,axiom,
    ! [A: rat,B: rat] : uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% minus_diff_eq
tff(fact_3043_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,minus_8373710615458151222nteger(A,B)) = minus_8373710615458151222nteger(B,A) ).

% minus_diff_eq
tff(fact_3044_power__add__numeral2,axiom,
    ! [A: assn,M2: num,N: num,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))),B)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3045_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3046_power__add__numeral2,axiom,
    ! [A: int,M2: num,N: num,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(M2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(N))),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3047_power__add__numeral2,axiom,
    ! [A: code_integer,M2: num,N: num,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,numeral_numeral_nat(M2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,numeral_numeral_nat(N))),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3048_power__add__numeral2,axiom,
    ! [A: code_natural,M2: num,N: num,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,numeral_numeral_nat(M2))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,numeral_numeral_nat(N))),B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,numeral_numeral_nat(plus_plus_num(M2,N)))),B) ).

% power_add_numeral2
tff(fact_3049_power__add__numeral,axiom,
    ! [A: assn,M2: num,N: num] : aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),numeral_numeral_nat(M2))),aa_nat_assn(power_power_assn(A),numeral_numeral_nat(N))) = aa_nat_assn(power_power_assn(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3050_power__add__numeral,axiom,
    ! [A: nat,M2: 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,A),numeral_numeral_nat(M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(N))) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3051_power__add__numeral,axiom,
    ! [A: int,M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),numeral_numeral_nat(M2))),aa_nat_int(power_power_int(A),numeral_numeral_nat(N))) = aa_nat_int(power_power_int(A),numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3052_power__add__numeral,axiom,
    ! [A: code_integer,M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,numeral_numeral_nat(M2))),power_8256067586552552935nteger(A,numeral_numeral_nat(N))) = power_8256067586552552935nteger(A,numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3053_power__add__numeral,axiom,
    ! [A: code_natural,M2: num,N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,numeral_numeral_nat(M2))),power_7079662738309270450atural(A,numeral_numeral_nat(N))) = power_7079662738309270450atural(A,numeral_numeral_nat(plus_plus_num(M2,N))) ).

% power_add_numeral
tff(fact_3054_diff__ge__0__iff__ge,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3055_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(ord_less_eq_rat(B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3056_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: 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,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% diff_ge_0_iff_ge
tff(fact_3057_diff__gt__0__iff__gt,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),minus_8373710615458151222nteger(A,B))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3058_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))
    <=> aa_rat_o(ord_less_rat(B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3059_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: 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,A),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ).

% diff_gt_0_iff_gt
tff(fact_3060_diff__add__zero,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = zero_zero_nat ).

% diff_add_zero
tff(fact_3061_diff__add__zero,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) = zero_z2226904508553997617atural ).

% diff_add_zero
tff(fact_3062_diff__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),one_one_rat) = zero_zero_rat ).

% diff_numeral_special(9)
tff(fact_3063_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_3064_diff__numeral__special_I9_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% diff_numeral_special(9)
tff(fact_3065_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)),numeral_numeral_nat(V2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(V2))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),numeral_numeral_nat(V2))) ).

% distrib_right_numeral
tff(fact_3066_distrib__right__numeral,axiom,
    ! [A: int,B: int,V2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),numeral_numeral_int(V2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V2))) ).

% distrib_right_numeral
tff(fact_3067_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),numeral_numeral_rat(V2)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V2))) ).

% distrib_right_numeral
tff(fact_3068_distrib__right__numeral,axiom,
    ! [A: code_integer,B: code_integer,V2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),numera6620942414471956472nteger(V2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),numera6620942414471956472nteger(V2))) ).

% distrib_right_numeral
tff(fact_3069_distrib__right__numeral,axiom,
    ! [A: code_natural,B: code_natural,V2: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)),numera5444537566228673987atural(V2)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),numera5444537566228673987atural(V2))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),numera5444537566228673987atural(V2))) ).

% distrib_right_numeral
tff(fact_3070_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V2)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V2)),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(V2)),C)) ).

% distrib_left_numeral
tff(fact_3071_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),C)) ).

% distrib_left_numeral
tff(fact_3072_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),C)) ).

% distrib_left_numeral
tff(fact_3073_distrib__left__numeral,axiom,
    ! [V2: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),C)) ).

% distrib_left_numeral
tff(fact_3074_distrib__left__numeral,axiom,
    ! [V2: num,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V2)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V2)),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(V2)),C)) ).

% distrib_left_numeral
tff(fact_3075_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),B)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_3076_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),B)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_3077_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),C)) ).

% right_diff_distrib_numeral
tff(fact_3078_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V2: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),numeral_numeral_int(V2)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(V2))),aa_int_int(aa_int_fun_int_int(times_times_int,B),numeral_numeral_int(V2))) ).

% left_diff_distrib_numeral
tff(fact_3079_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V2: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),numeral_numeral_rat(V2)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),numeral_numeral_rat(V2))) ).

% left_diff_distrib_numeral
tff(fact_3080_left__diff__distrib__numeral,axiom,
    ! [A: code_integer,B: code_integer,V2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),numera6620942414471956472nteger(V2)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),numera6620942414471956472nteger(V2))) ).

% left_diff_distrib_numeral
tff(fact_3081_diff__0,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,zero_zero_int),A) = aa_int_int(uminus_uminus_int,A) ).

% diff_0
tff(fact_3082_diff__0,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,zero_zero_rat),A) = uminus_uminus_rat(A) ).

% diff_0
tff(fact_3083_diff__0,axiom,
    ! [A: code_integer] : minus_8373710615458151222nteger(zero_z3403309356797280102nteger,A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ).

% diff_0
tff(fact_3084_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = numeral_numeral_int(times_times_num(M2,N)) ).

% mult_neg_numeral_simps(1)
tff(fact_3085_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(times_times_num(M2,N)) ).

% mult_neg_numeral_simps(1)
tff(fact_3086_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(times_times_num(M2,N)) ).

% mult_neg_numeral_simps(1)
tff(fact_3087_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(2)
tff(fact_3088_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(2)
tff(fact_3089_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(2)
tff(fact_3090_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(3)
tff(fact_3091_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(M2)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(3)
tff(fact_3092_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(M2,N))) ).

% mult_neg_numeral_simps(3)
tff(fact_3093_diff__minus__eq__add,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(uminus_uminus_int,B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) ).

% diff_minus_eq_add
tff(fact_3094_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),uminus_uminus_rat(B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) ).

% diff_minus_eq_add
tff(fact_3095_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) ).

% diff_minus_eq_add
tff(fact_3096_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,A)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A) ).

% uminus_add_conv_diff
tff(fact_3097_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(A)),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A) ).

% uminus_add_conv_diff
tff(fact_3098_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) = minus_8373710615458151222nteger(B,A) ).

% uminus_add_conv_diff
tff(fact_3099_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W))),A)
    <=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_le_eq_numeral1(1)
tff(fact_3100_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)))
    <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% le_divide_eq_numeral1(1)
tff(fact_3101_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)) = A )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_3102_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)) )
    <=> $ite(numeral_numeral_rat(W) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W)) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_3103_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_3104_diff__numeral__special_I12_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(one_one_rat)) = zero_zero_rat ).

% diff_numeral_special(12)
tff(fact_3105_diff__numeral__special_I12_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% diff_numeral_special(12)
tff(fact_3106_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W))),A)
    <=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))) ) ).

% divide_less_eq_numeral1(1)
tff(fact_3107_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),numeral_numeral_rat(W)))
    <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(W))),B) ) ).

% less_divide_eq_numeral1(1)
tff(fact_3108_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3109_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3110_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W))) = A )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W))),A = zero_zero_rat) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3111_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W))) )
    <=> $ite(uminus_uminus_rat(numeral_numeral_rat(W)) != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W))) = B,A = zero_zero_rat) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3112_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W)))),A)
    <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))),B) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3113_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),uminus_uminus_rat(numeral_numeral_rat(W))))
    <=> aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3114_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( ( A = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_3115_diff__eq__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = minus_8373710615458151222nteger(C,D2) )
     => ( ( A = B )
      <=> ( C = D2 ) ) ) ).

% diff_eq_diff_eq
tff(fact_3116_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),C)),B) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3117_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3118_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : minus_8373710615458151222nteger(minus_8373710615458151222nteger(A,C),B) = minus_8373710615458151222nteger(minus_8373710615458151222nteger(A,B),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3119_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : minus_7197305767214868737atural(minus_7197305767214868737atural(A,C),B) = minus_7197305767214868737atural(minus_7197305767214868737atural(A,B),C) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_3120_diff__mono,axiom,
    ! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(D2),C)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,D2)) ) ) ).

% diff_mono
tff(fact_3121_diff__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(D2),C)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_mono
tff(fact_3122_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_mono
tff(fact_3123_diff__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),A)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(C,A)),minus_8373710615458151222nteger(C,B)) ) ).

% diff_left_mono
tff(fact_3124_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_left_mono
tff(fact_3125_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_left_mono
tff(fact_3126_diff__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,C)) ) ).

% diff_right_mono
tff(fact_3127_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_right_mono
tff(fact_3128_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_right_mono
tff(fact_3129_diff__eq__diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = minus_8373710615458151222nteger(C,D2) )
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
      <=> aa_Code_integer_o(ord_le3102999989581377725nteger(C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3130_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D2) )
     => ( aa_rat_o(ord_less_eq_rat(A),B)
      <=> aa_rat_o(ord_less_eq_rat(C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3131_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),D2) ) ) ).

% diff_eq_diff_less_eq
tff(fact_3132_eq__iff__diff__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = zero_zero_rat ) ) ).

% eq_iff_diff_eq_0
tff(fact_3133_eq__iff__diff__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
    <=> ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = zero_zero_int ) ) ).

% eq_iff_diff_eq_0
tff(fact_3134_eq__iff__diff__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A = B )
    <=> ( minus_8373710615458151222nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% eq_iff_diff_eq_0
tff(fact_3135_diff__strict__mono,axiom,
    ! [A: code_integer,B: code_integer,D2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(D2),C)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,D2)) ) ) ).

% diff_strict_mono
tff(fact_3136_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D2: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_o(ord_less_rat(D2),C)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_3137_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,D2),C)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)) ) ) ).

% diff_strict_mono
tff(fact_3138_diff__eq__diff__less,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = minus_8373710615458151222nteger(C,D2) )
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
      <=> aa_Code_integer_o(ord_le6747313008572928689nteger(C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_3139_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),D2) )
     => ( aa_rat_o(ord_less_rat(A),B)
      <=> aa_rat_o(ord_less_rat(C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_3140_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),D2) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,C),D2) ) ) ).

% diff_eq_diff_less
tff(fact_3141_diff__strict__left__mono,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(minus_8373710615458151222nteger(C,A)),minus_8373710615458151222nteger(C,B)) ) ).

% diff_strict_left_mono
tff(fact_3142_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B)) ) ).

% diff_strict_left_mono
tff(fact_3143_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,C),A)),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B)) ) ).

% diff_strict_left_mono
tff(fact_3144_diff__strict__right__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(minus_8373710615458151222nteger(A,C)),minus_8373710615458151222nteger(B,C)) ) ).

% diff_strict_right_mono
tff(fact_3145_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) ) ).

% diff_strict_right_mono
tff(fact_3146_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) ) ).

% diff_strict_right_mono
tff(fact_3147_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),A) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A2),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3148_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),A) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A2),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub1
tff(fact_3149_group__cancel_Osub1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer,B: code_integer] :
      ( ( A2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),A) )
     => ( minus_8373710615458151222nteger(A2,B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub1
tff(fact_3150_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = C )
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) ) ) ).

% diff_eq_eq
tff(fact_3151_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = C )
    <=> ( A = aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) ) ) ).

% diff_eq_eq
tff(fact_3152_diff__eq__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( minus_8373710615458151222nteger(A,B) = C )
    <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) ) ) ).

% diff_eq_eq
tff(fact_3153_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3154_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B) )
    <=> ( aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3155_eq__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( A = minus_8373710615458151222nteger(C,B) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B) = C ) ) ).

% eq_diff_eq
tff(fact_3156_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ).

% add_diff_eq
tff(fact_3157_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ).

% add_diff_eq
tff(fact_3158_add__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),C) ).

% add_diff_eq
tff(fact_3159_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_diff_eq2
tff(fact_3160_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_diff_eq2
tff(fact_3161_diff__diff__eq2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : minus_8373710615458151222nteger(A,minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C),B) ).

% diff_diff_eq2
tff(fact_3162_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),C)),B) ).

% diff_add_eq
tff(fact_3163_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C)),B) ).

% diff_add_eq
tff(fact_3164_diff__add__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C),B) ).

% diff_add_eq
tff(fact_3165_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3166_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),C)),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3167_diff__add__eq__diff__diff__swap,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) = minus_8373710615458151222nteger(minus_8373710615458151222nteger(A,C),B) ).

% diff_add_eq_diff_diff_swap
tff(fact_3168_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B) = A )
     => ( C = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) ) ) ).

% add_implies_diff
tff(fact_3169_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B) = A )
     => ( C = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B) ) ) ).

% add_implies_diff
tff(fact_3170_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B) = A )
     => ( C = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) ) ) ).

% add_implies_diff
tff(fact_3171_add__implies__diff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B) = A )
     => ( C = minus_8373710615458151222nteger(A,B) ) ) ).

% add_implies_diff
tff(fact_3172_add__implies__diff,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,C),B) = A )
     => ( C = minus_7197305767214868737atural(A,B) ) ) ).

% add_implies_diff
tff(fact_3173_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),C)) ).

% diff_diff_eq
tff(fact_3174_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)) ).

% diff_diff_eq
tff(fact_3175_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),C)) ).

% diff_diff_eq
tff(fact_3176_diff__diff__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : minus_8373710615458151222nteger(minus_8373710615458151222nteger(A,B),C) = minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),C)) ).

% diff_diff_eq
tff(fact_3177_diff__diff__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : minus_7197305767214868737atural(minus_7197305767214868737atural(A,B),C) = minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),C)) ).

% diff_diff_eq
tff(fact_3178_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)) ).

% right_diff_distrib'
tff(fact_3179_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib'
tff(fact_3180_right__diff__distrib_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% right_diff_distrib'
tff(fact_3181_right__diff__distrib_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),minus_7197305767214868737atural(B,C)) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)) ).

% right_diff_distrib'
tff(fact_3182_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)) ).

% left_diff_distrib'
tff(fact_3183_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)) ).

% left_diff_distrib'
tff(fact_3184_left__diff__distrib_H,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(B,C)),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)) ).

% left_diff_distrib'
tff(fact_3185_left__diff__distrib_H,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,minus_7197305767214868737atural(B,C)),A) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)) ).

% left_diff_distrib'
tff(fact_3186_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ).

% right_diff_distrib
tff(fact_3187_right__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ).

% right_diff_distrib
tff(fact_3188_left__diff__distrib,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ).

% left_diff_distrib
tff(fact_3189_left__diff__distrib,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ).

% left_diff_distrib
tff(fact_3190_minus__diff__commute,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,B)),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),B) ).

% minus_diff_commute
tff(fact_3191_minus__diff__commute,axiom,
    ! [B: rat,A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(B)),A) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(A)),B) ).

% minus_diff_commute
tff(fact_3192_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),B) ).

% minus_diff_commute
tff(fact_3193_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: int,A: int,B: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3194_left__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3195_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3196_left__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3197_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: int,A: int,B: int] :
      ( aa_int_o(nO_MATCH_rat_int(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3198_left__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT243659986023775650nteger(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3199_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C: int,A: int,B: int] :
      ( aa_int_o(nO_MAT4469370267861770177al_int(divide5121882707175180666atural(X,Y)),C)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3200_left__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(nO_MAT2693162092599435762nteger(divide5121882707175180666atural(X,Y)),C)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),C) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ) ).

% left_diff_distrib_NO_MATCH
tff(fact_3201_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: int,B: int,C: int] :
      ( aa_int_o(aa_nat_fun_int_o(nO_MATCH_nat_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3202_right__diff__distrib__NO__MATCH,axiom,
    ! [X: nat,Y: nat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT8694959213317031834nteger(aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3203_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(nO_MATCH_int_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3204_right__diff__distrib__NO__MATCH,axiom,
    ! [X: int,Y: int,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT9066612773553876470nteger(aa_int_int(aa_int_fun_int_int(divide_divide_int,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3205_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: int,B: int,C: int] :
      ( aa_int_o(nO_MATCH_rat_int(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3206_right__diff__distrib__NO__MATCH,axiom,
    ! [X: rat,Y: rat,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT243659986023775650nteger(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3207_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: int,B: int,C: int] :
      ( aa_int_o(nO_MAT4469370267861770177al_int(divide5121882707175180666atural(X,Y)),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,B),C)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3208_right__diff__distrib__NO__MATCH,axiom,
    ! [X: code_natural,Y: code_natural,A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(nO_MAT2693162092599435762nteger(divide5121882707175180666atural(X,Y)),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),minus_8373710615458151222nteger(B,C)) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)) ) ) ).

% right_diff_distrib_NO_MATCH
tff(fact_3209_le__iff__diff__le__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(A,B)),zero_z3403309356797280102nteger) ) ).

% le_iff_diff_le_0
tff(fact_3210_le__iff__diff__le__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
    <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% le_iff_diff_le_0
tff(fact_3211_le__iff__diff__le__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% le_iff_diff_le_0
tff(fact_3212_less__iff__diff__less__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(minus_8373710615458151222nteger(A,B)),zero_z3403309356797280102nteger) ) ).

% less_iff_diff_less_0
tff(fact_3213_less__iff__diff__less__0,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
    <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),zero_zero_rat) ) ).

% less_iff_diff_less_0
tff(fact_3214_less__iff__diff__less__0,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)),zero_zero_int) ) ).

% less_iff_diff_less_0
tff(fact_3215_diff__le__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(A,B)),C)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_le_eq
tff(fact_3216_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_le_eq
tff(fact_3217_diff__le__eq,axiom,
    ! [A: int,B: 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,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_le_eq
tff(fact_3218_le__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).

% le_diff_eq
tff(fact_3219_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% le_diff_eq
tff(fact_3220_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% le_diff_eq
tff(fact_3221_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),A) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_3222_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => 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,B),C)),A)) ) ).

% le_add_diff
tff(fact_3223_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_3224_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_3225_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),B)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_3226_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_3227_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)),C) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_3228_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A)),B) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_3229_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A)) = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_3230_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,C),A) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_3231_less__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),minus_8373710615458151222nteger(C,B))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)),C) ) ).

% less_diff_eq
tff(fact_3232_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,C),B))
    <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),C) ) ).

% less_diff_eq
tff(fact_3233_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(minus_minus_int,C),B))
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)),C) ) ).

% less_diff_eq
tff(fact_3234_diff__less__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(minus_8373710615458151222nteger(A,B)),C)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),B)) ) ).

% diff_less_eq
tff(fact_3235_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),C)
    <=> aa_rat_o(ord_less_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),B)) ) ).

% diff_less_eq
tff(fact_3236_diff__less__eq,axiom,
    ! [A: int,B: 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,A),B)),C)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),B)) ) ).

% diff_less_eq
tff(fact_3237_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2) )
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3238_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( 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,A),B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3239_eq__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),E2)),C) = D2 ) ) ).

% eq_add_iff1
tff(fact_3240_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2) )
    <=> ( C = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_3241_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: int] :
      ( ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),E2)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2) )
    <=> ( 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,B),A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_3242_eq__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2) )
    <=> ( C = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(B,A)),E2)),D2) ) ) ).

% eq_add_iff2
tff(fact_3243_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y)) ).

% square_diff_square_factored
tff(fact_3244_square__diff__square__factored,axiom,
    ! [X: 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,X),X)),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,X),Y)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)) ).

% square_diff_square_factored
tff(fact_3245_square__diff__square__factored,axiom,
    ! [X: code_integer,Y: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y)),minus_8373710615458151222nteger(X,Y)) ).

% square_diff_square_factored
tff(fact_3246_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3247_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3248_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3249_diff__conv__add__uminus,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(uminus_uminus_int,B)) ).

% diff_conv_add_uminus
tff(fact_3250_diff__conv__add__uminus,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),uminus_uminus_rat(B)) ).

% diff_conv_add_uminus
tff(fact_3251_diff__conv__add__uminus,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B)) ).

% diff_conv_add_uminus
tff(fact_3252_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2 = aa_int_int(aa_int_fun_int_int(plus_plus_int,K),B) )
     => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B2) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,K)),aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_3253_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2 = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,K),B) )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B2) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(K)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)) ) ) ).

% group_cancel.sub2
tff(fact_3254_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2 = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,K),B) )
     => ( minus_8373710615458151222nteger(A,B2) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K)),minus_8373710615458151222nteger(A,B)) ) ) ).

% group_cancel.sub2
tff(fact_3255_diff__eq,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,X),Y) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,X),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,Y)) ).

% diff_eq
tff(fact_3256_diff__eq,axiom,
    ! [X: set_nat,Y: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,X),Y) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,X),aa_set_nat_set_nat(uminus5710092332889474511et_nat,Y)) ).

% diff_eq
tff(fact_3257_diff__eq,axiom,
    ! [X: fun_Pr3238374608701176162_int_o,Y: fun_Pr3238374608701176162_int_o] : aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(minus_827889751674334651_int_o,X),Y) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,X),aa_fun5349570537165229698_int_o(uminus103585356917985579_int_o,Y)) ).

% diff_eq
tff(fact_3258_diff__eq,axiom,
    ! [X: assn,Y: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,X),Y) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,X),aa_assn_assn(uminus_uminus_assn,Y)) ).

% diff_eq
tff(fact_3259_one__le__numeral,axiom,
    ! [N: num] : aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),numera6620942414471956472nteger(N)) ).

% one_le_numeral
tff(fact_3260_one__le__numeral,axiom,
    ! [N: num] : aa_Code_natural_o(ord_le1926595141338095240atural(one_one_Code_natural),numera5444537566228673987atural(N)) ).

% one_le_numeral
tff(fact_3261_one__le__numeral,axiom,
    ! [N: num] : aa_rat_o(ord_less_eq_rat(one_one_rat),numeral_numeral_rat(N)) ).

% one_le_numeral
tff(fact_3262_one__le__numeral,axiom,
    ! [N: num] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),numeral_numeral_nat(N)) ).

% one_le_numeral
tff(fact_3263_one__le__numeral,axiom,
    ! [N: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),numeral_numeral_int(N)) ).

% one_le_numeral
tff(fact_3264_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(N)),one_one_nat) ).

% not_numeral_less_one
tff(fact_3265_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(N)),one_one_int) ).

% not_numeral_less_one
tff(fact_3266_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_rat_o(ord_less_rat(numeral_numeral_rat(N)),one_one_rat) ).

% not_numeral_less_one
tff(fact_3267_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(numera6620942414471956472nteger(N)),one_one_Code_integer) ).

% not_numeral_less_one
tff(fact_3268_not__numeral__less__one,axiom,
    ! [N: num] : ~ aa_Code_natural_o(ord_le5570908160329646204atural(numera5444537566228673987atural(N)),one_one_Code_natural) ).

% not_numeral_less_one
tff(fact_3269_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ).

% one_plus_numeral_commute
tff(fact_3270_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).

% one_plus_numeral_commute
tff(fact_3271_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).

% one_plus_numeral_commute
tff(fact_3272_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),numera6620942414471956472nteger(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(X)),one_one_Code_integer) ).

% one_plus_numeral_commute
tff(fact_3273_one__plus__numeral__commute,axiom,
    ! [X: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),numera5444537566228673987atural(X)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(X)),one_one_Code_natural) ).

% one_plus_numeral_commute
tff(fact_3274_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_int != aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) ).

% one_neq_neg_numeral
tff(fact_3275_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_rat != uminus_uminus_rat(numeral_numeral_rat(N)) ).

% one_neq_neg_numeral
tff(fact_3276_one__neq__neg__numeral,axiom,
    ! [N: num] : one_one_Code_integer != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) ).

% one_neq_neg_numeral
tff(fact_3277_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_int(N) != aa_int_int(uminus_uminus_int,one_one_int) ).

% numeral_neq_neg_one
tff(fact_3278_numeral__neq__neg__one,axiom,
    ! [N: num] : numeral_numeral_rat(N) != uminus_uminus_rat(one_one_rat) ).

% numeral_neq_neg_one
tff(fact_3279_numeral__neq__neg__one,axiom,
    ! [N: num] : numera6620942414471956472nteger(N) != aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% numeral_neq_neg_one
tff(fact_3280_le__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_3281_le__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_3282_le__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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,A),B)),E2)),C)),D2) ) ).

% le_add_iff1
tff(fact_3283_le__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(B,A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_3284_le__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_3285_le__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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,B),A)),E2)),D2)) ) ).

% le_add_iff2
tff(fact_3286_less__add__iff1,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(A,B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_3287_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_3288_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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,A),B)),E2)),C)),D2) ) ).

% less_add_iff1
tff(fact_3289_less__add__iff2,axiom,
    ! [A: code_integer,E2: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),E2)),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),E2)),D2))
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,minus_8373710615458151222nteger(B,A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_3290_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),E2)),C)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),E2)),D2))
    <=> aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_3291_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D2: 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,A),E2)),C)),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),E2)),D2))
    <=> 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,B),A)),E2)),D2)) ) ).

% less_add_iff2
tff(fact_3292_square__diff__one__factored,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),X)),one_one_rat) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),one_one_rat)),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),one_one_rat)) ).

% square_diff_one_factored
tff(fact_3293_square__diff__one__factored,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,X),X)),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,X),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,X),one_one_int)) ).

% square_diff_one_factored
tff(fact_3294_square__diff__one__factored,axiom,
    ! [X: code_integer] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),one_one_Code_integer)),minus_8373710615458151222nteger(X,one_one_Code_integer)) ).

% square_diff_one_factored
tff(fact_3295_divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z)),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% divide_diff_eq_iff
tff(fact_3296_diff__divide__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),Y)),Z) ) ) ).

% diff_divide_eq_iff
tff(fact_3297_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z)) ) ) ) ).

% diff_frac_eq
tff(fact_3298_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: rat,B: rat,Z: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),Z)) = $ite(Z = zero_zero_rat,A,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),Z)),B)),Z)) ).

% add_divide_eq_if_simps(4)
tff(fact_3299_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(ord_le3102999989581377725nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_one_le_neg_numeral
tff(fact_3300_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ aa_rat_o(ord_less_eq_rat(one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M2))) ).

% not_one_le_neg_numeral
tff(fact_3301_not__one__le__neg__numeral,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) ).

% not_one_le_neg_numeral
tff(fact_3302_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(ord_le3102999989581377725nteger(numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_le_neg_one
tff(fact_3303_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(M2)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_le_neg_one
tff(fact_3304_not__numeral__le__neg__one,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_le_neg_one
tff(fact_3305_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% neg_numeral_le_neg_one
tff(fact_3306_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(numeral_numeral_rat(M2))),uminus_uminus_rat(one_one_rat)) ).

% neg_numeral_le_neg_one
tff(fact_3307_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) ).

% neg_numeral_le_neg_one
tff(fact_3308_neg__one__le__numeral,axiom,
    ! [M2: num] : aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).

% neg_one_le_numeral
tff(fact_3309_neg__one__le__numeral,axiom,
    ! [M2: num] : aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M2)) ).

% neg_one_le_numeral
tff(fact_3310_neg__one__le__numeral,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M2)) ).

% neg_one_le_numeral
tff(fact_3311_neg__numeral__le__one,axiom,
    ! [M2: num] : aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).

% neg_numeral_le_one
tff(fact_3312_neg__numeral__le__one,axiom,
    ! [M2: num] : aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(numeral_numeral_rat(M2))),one_one_rat) ).

% neg_numeral_le_one
tff(fact_3313_neg__numeral__le__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) ).

% neg_numeral_le_one
tff(fact_3314_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = numeral_numeral_rat(W) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C),numeral_numeral_rat(W) = zero_zero_rat) ) ).

% divide_eq_eq_numeral(1)
tff(fact_3315_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( numeral_numeral_rat(W) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C) = B,numeral_numeral_rat(W) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(1)
tff(fact_3316_not__neg__one__less__neg__numeral,axiom,
    ! [M2: 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,numeral_numeral_int(M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_3317_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_rat_o(ord_less_rat(uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_3318_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_neg_one_less_neg_numeral
tff(fact_3319_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) ).

% not_one_less_neg_numeral
tff(fact_3320_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_rat_o(ord_less_rat(one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M2))) ).

% not_one_less_neg_numeral
tff(fact_3321_not__one__less__neg__numeral,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) ).

% not_one_less_neg_numeral
tff(fact_3322_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) ).

% not_numeral_less_neg_one
tff(fact_3323_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_rat_o(ord_less_rat(numeral_numeral_rat(M2)),uminus_uminus_rat(one_one_rat)) ).

% not_numeral_less_neg_one
tff(fact_3324_not__numeral__less__neg__one,axiom,
    ! [M2: num] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% not_numeral_less_neg_one
tff(fact_3325_neg__one__less__numeral,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(M2)) ).

% neg_one_less_numeral
tff(fact_3326_neg__one__less__numeral,axiom,
    ! [M2: num] : aa_rat_o(ord_less_rat(uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(M2)) ).

% neg_one_less_numeral
tff(fact_3327_neg__one__less__numeral,axiom,
    ! [M2: num] : aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(M2)) ).

% neg_one_less_numeral
tff(fact_3328_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) ).

% neg_numeral_less_one
tff(fact_3329_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_rat_o(ord_less_rat(uminus_uminus_rat(numeral_numeral_rat(M2))),one_one_rat) ).

% neg_numeral_less_one
tff(fact_3330_neg__numeral__less__one,axiom,
    ! [M2: num] : aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) ).

% neg_numeral_less_one
tff(fact_3331_dbl__dec__def,axiom,
    ! [X: rat] : neg_nu3179335615603231917ec_rat(X) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),X)),one_one_rat) ).

% dbl_dec_def
tff(fact_3332_dbl__dec__def,axiom,
    ! [X: int] : aa_int_int(neg_nu3811975205180677377ec_int,X) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),X)),one_one_int) ).

% dbl_dec_def
tff(fact_3333_dbl__dec__def,axiom,
    ! [X: code_integer] : neg_nu7757733837767384882nteger(X) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),X),one_one_Code_integer) ).

% dbl_dec_def
tff(fact_3334_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_le_eq
tff(fact_3335_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,W),Z))
        <=> aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,W),Y))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),zero_zero_rat) ) ) ) ).

% frac_less_eq
tff(fact_3336_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,X),Z))),Y) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(X)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),Z))),Z) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_3337_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z)),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(5)
tff(fact_3338_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: rat,Z: rat,B: rat] :
      aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,A),Z))),B) = $ite(Z = zero_zero_rat,uminus_uminus_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),Z))),Z)) ).

% add_divide_eq_if_simps(6)
tff(fact_3339_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] : aa_Pro8055486230010607173sn_nat(count_2530312006313534765n_assn(nil_Pr5671120429643327159n_assn),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3340_count__list_Osimps_I1_J,axiom,
    ! [Y: int] : aa_int_nat(count_list_int(nil_int),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3341_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] : aa_nat_nat(count_list_nat(nil_nat),Y) = zero_zero_nat ).

% count_list.simps(1)
tff(fact_3342_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_3343_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(numeral_numeral_rat(W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(ord_less_rat(numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_3344_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) = uminus_uminus_rat(numeral_numeral_rat(W)) )
    <=> $ite(C != zero_zero_rat,B = aa_rat_rat(aa_rat_fun_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_3345_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( uminus_uminus_rat(numeral_numeral_rat(W)) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C) )
    <=> $ite(C != zero_zero_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C) = B,uminus_uminus_rat(numeral_numeral_rat(W)) = zero_zero_rat) ) ).

% eq_divide_eq_numeral(2)
tff(fact_3346_mult__eq__if,axiom,
    ! [M2: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N) = $ite(M2 = 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,M2),one_one_nat)),N))) ).

% mult_eq_if
tff(fact_3347_scaling__mono,axiom,
    ! [U: rat,V2: rat,R2: rat,S3: rat] :
      ( aa_rat_o(ord_less_eq_rat(U),V2)
     => ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),R2)
       => ( aa_rat_o(ord_less_eq_rat(R2),S3)
         => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,U),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,R2),aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,V2),U))),S3))),V2) ) ) ) ).

% scaling_mono
tff(fact_3348_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),numeral_numeral_rat(W))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),aa_rat_o(ord_less_eq_rat(zero_zero_rat),numeral_numeral_rat(W))) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_3349_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(W)),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),C)),aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(W)),zero_zero_rat)) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_3350_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(numeral_numeral_rat(W))),aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),aa_rat_o(ord_less_rat(uminus_uminus_rat(numeral_numeral_rat(W))),zero_zero_rat)) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_3351_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,B),C)),uminus_uminus_rat(numeral_numeral_rat(W)))
    <=> $ite(
          aa_rat_o(ord_less_rat(zero_zero_rat),C),
          aa_rat_o(ord_less_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),
          $ite(aa_rat_o(ord_less_rat(C),zero_zero_rat),aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),C)),B),aa_rat_o(ord_less_rat(zero_zero_rat),uminus_uminus_rat(numeral_numeral_rat(W)))) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_3352_power__diff__power__eq,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( divide6298287555418463151nteger(power_8256067586552552935nteger(A,M2),power_8256067586552552935nteger(A,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),power_8256067586552552935nteger(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_3353_power__diff__power__eq,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( A != 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,A),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),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,A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_3354_power__diff__power__eq,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(A),M2)),aa_nat_int(power_power_int(A),N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_3355_power__diff__power__eq,axiom,
    ! [A: code_natural,M2: nat,N: nat] :
      ( ( A != zero_z2226904508553997617atural )
     => ( divide5121882707175180666atural(power_7079662738309270450atural(A,M2),power_7079662738309270450atural(A,N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2),power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N)),divide5121882707175180666atural(one_one_Code_natural,power_7079662738309270450atural(A,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ).

% power_diff_power_eq
tff(fact_3356_power__eq__if,axiom,
    ! [P4: rat,M2: nat] :
      power_power_rat(P4,M2) = $ite(M2 = zero_zero_nat,one_one_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,P4),power_power_rat(P4,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3357_power__eq__if,axiom,
    ! [P4: assn,M2: nat] :
      aa_nat_assn(power_power_assn(P4),M2) = $ite(M2 = zero_zero_nat,one_one_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,P4),aa_nat_assn(power_power_assn(P4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3358_power__eq__if,axiom,
    ! [P4: nat,M2: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P4),M2) = $ite(M2 = zero_zero_nat,one_one_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,P4),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,P4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3359_power__eq__if,axiom,
    ! [P4: int,M2: nat] :
      aa_nat_int(power_power_int(P4),M2) = $ite(M2 = zero_zero_nat,one_one_int,aa_int_int(aa_int_fun_int_int(times_times_int,P4),aa_nat_int(power_power_int(P4),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3360_power__eq__if,axiom,
    ! [P4: code_integer,M2: nat] :
      power_8256067586552552935nteger(P4,M2) = $ite(M2 = zero_zero_nat,one_one_Code_integer,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,P4),power_8256067586552552935nteger(P4,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3361_power__eq__if,axiom,
    ! [P4: code_natural,M2: nat] :
      power_7079662738309270450atural(P4,M2) = $ite(M2 = zero_zero_nat,one_one_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,P4),power_7079662738309270450atural(P4,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))) ).

% power_eq_if
tff(fact_3362_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V2))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(V2,W)))),Y) ).

% semiring_norm(170)
tff(fact_3363_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(times_times_num(V2,W)))),Y) ).

% semiring_norm(170)
tff(fact_3364_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(V2,W)))),Y) ).

% semiring_norm(170)
tff(fact_3365_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(V2)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(times_times_num(V2,W)))),Y) ).

% semiring_norm(171)
tff(fact_3366_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(V2)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(times_times_num(V2,W)))),Y) ).

% semiring_norm(171)
tff(fact_3367_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(V2)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(times_times_num(V2,W)))),Y) ).

% semiring_norm(171)
tff(fact_3368_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(V2))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(W))),Y)) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(times_times_num(V2,W))),Y) ).

% semiring_norm(172)
tff(fact_3369_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(V2))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(W))),Y)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(times_times_num(V2,W))),Y) ).

% semiring_norm(172)
tff(fact_3370_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(V2))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))),Y)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(times_times_num(V2,W))),Y) ).

% semiring_norm(172)
tff(fact_3371_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_3372_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)
     => ( power_power_rat(uminus_uminus_rat(one_one_rat),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = 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_3373_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)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),K)) = 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_3374_numeral__times__minus__swap,axiom,
    ! [W: num,X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(W)),aa_int_int(uminus_uminus_int,X)) = aa_int_int(aa_int_fun_int_int(times_times_int,X),aa_int_int(uminus_uminus_int,numeral_numeral_int(W))) ).

% numeral_times_minus_swap
tff(fact_3375_numeral__times__minus__swap,axiom,
    ! [W: num,X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(W)),uminus_uminus_rat(X)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,X),uminus_uminus_rat(numeral_numeral_rat(W))) ).

% numeral_times_minus_swap
tff(fact_3376_numeral__times__minus__swap,axiom,
    ! [W: num,X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(W)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(W))) ).

% numeral_times_minus_swap
tff(fact_3377_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      slice_int(Begin,End,cons_int(X,Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        cons_int(X,slice_int(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_int(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_3378_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      slice_nat(Begin,End,cons_nat(X,Xs)) = $ite(
        ( ( Begin = zero_zero_nat )
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),End) ),
        cons_nat(X,slice_nat(Begin,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs)),
        slice_nat(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Begin),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,End),one_one_nat),Xs) ) ).

% slice_Cons
tff(fact_3379_inf__period_I1_J,axiom,
    ! [Pa: fun_int_o,D: int,Qa: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(Pa,X2)
        <=> aa_int_o(Pa,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))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Qa,X2)
          <=> aa_int_o(Qa,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))) )
       => ! [X3: int,K3: int] :
            ( ( aa_int_o(Pa,X3)
              & aa_int_o(Qa,X3) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))
              & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_3380_inf__period_I1_J,axiom,
    ! [Pa: fun_Code_integer_o,D: code_integer,Qa: fun_Code_integer_o] :
      ( ! [X2: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(Pa,X2)
        <=> aa_Code_integer_o(Pa,minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))) )
     => ( ! [X2: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Qa,X2)
          <=> aa_Code_integer_o(Qa,minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))) )
       => ! [X3: code_integer,K3: code_integer] :
            ( ( aa_Code_integer_o(Pa,X3)
              & aa_Code_integer_o(Qa,X3) )
          <=> ( aa_Code_integer_o(Pa,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D)))
              & aa_Code_integer_o(Qa,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D))) ) ) ) ) ).

% inf_period(1)
tff(fact_3381_inf__period_I2_J,axiom,
    ! [Pa: fun_int_o,D: int,Qa: fun_int_o] :
      ( ! [X2: int,K2: int] :
          ( aa_int_o(Pa,X2)
        <=> aa_int_o(Pa,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))) )
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Qa,X2)
          <=> aa_int_o(Qa,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))) )
       => ! [X3: int,K3: int] :
            ( ( aa_int_o(Pa,X3)
              | aa_int_o(Qa,X3) )
          <=> ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D)))
              | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_3382_inf__period_I2_J,axiom,
    ! [Pa: fun_Code_integer_o,D: code_integer,Qa: fun_Code_integer_o] :
      ( ! [X2: code_integer,K2: code_integer] :
          ( aa_Code_integer_o(Pa,X2)
        <=> aa_Code_integer_o(Pa,minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))) )
     => ( ! [X2: code_integer,K2: code_integer] :
            ( aa_Code_integer_o(Qa,X2)
          <=> aa_Code_integer_o(Qa,minus_8373710615458151222nteger(X2,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K2),D))) )
       => ! [X3: code_integer,K3: code_integer] :
            ( ( aa_Code_integer_o(Pa,X3)
              | aa_Code_integer_o(Qa,X3) )
          <=> ( aa_Code_integer_o(Pa,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D)))
              | aa_Code_integer_o(Qa,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D))) ) ) ) ) ).

% inf_period(2)
tff(fact_3383_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(N))) ).

% diff_numeral_special(5)
tff(fact_3384_diff__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(N)) = uminus_uminus_rat(numeral_numeral_rat(inc(N))) ).

% diff_numeral_special(5)
tff(fact_3385_diff__numeral__special_I5_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).

% diff_numeral_special(5)
tff(fact_3386_Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),A2)) = bot_bo2099793752762293965at_nat ).

% Diff_disjoint
tff(fact_3387_Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),A2)) = bot_bot_set_nat ).

% Diff_disjoint
tff(fact_3388_Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),minus_minus_set_int(B2,A2)) = bot_bot_set_int ).

% Diff_disjoint
tff(fact_3389_inter__compl__diff__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) ).

% inter_compl_diff_conv
tff(fact_3390_inter__compl__diff__conv,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) ).

% inter_compl_diff_conv
tff(fact_3391_Diff__Compl,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(uminus6524753893492686040at_nat,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% Diff_Compl
tff(fact_3392_Diff__Compl,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(uminus5710092332889474511et_nat,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Diff_Compl
tff(fact_3393_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_7964966981652229398n_assn(Begin,End,nil_Pr5671120429643327159n_assn) = nil_Pr5671120429643327159n_assn ).

% slice_Nil
tff(fact_3394_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_int(Begin,End,nil_int) = nil_int ).

% slice_Nil
tff(fact_3395_slice__Nil,axiom,
    ! [Begin: nat,End: nat] : slice_nat(Begin,End,nil_nat) = nil_nat ).

% slice_Nil
tff(fact_3396_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_P8527749157015355191n_assn] : slice_7964966981652229398n_assn(I2,I2,Xs) = nil_Pr5671120429643327159n_assn ).

% slice_eq_bounds_empty
tff(fact_3397_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_int] : slice_int(I2,I2,Xs) = nil_int ).

% slice_eq_bounds_empty
tff(fact_3398_slice__eq__bounds__empty,axiom,
    ! [I2: nat,Xs: list_nat] : slice_nat(I2,I2,Xs) = nil_nat ).

% slice_eq_bounds_empty
tff(fact_3399_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(M2))) ).

% add_neg_numeral_special(6)
tff(fact_3400_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(inc(M2))) ).

% add_neg_numeral_special(6)
tff(fact_3401_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(M2))) ).

% add_neg_numeral_special(6)
tff(fact_3402_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : 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,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_3403_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(N))) = uminus_uminus_rat(numeral_numeral_rat(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_3404_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(inc(N))) ).

% add_neg_numeral_special(5)
tff(fact_3405_diff__numeral__special_I6_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) = numeral_numeral_int(inc(M2)) ).

% diff_numeral_special(6)
tff(fact_3406_diff__numeral__special_I6_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(M2)),uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(inc(M2)) ).

% diff_numeral_special(6)
tff(fact_3407_diff__numeral__special_I6_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(inc(M2)) ).

% diff_numeral_special(6)
tff(fact_3408_Int__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,B2),C2)) ).

% Int_Diff
tff(fact_3409_Int__Diff,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,B2),C2)) ).

% Int_Diff
tff(fact_3410_Diff__Int2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),B2) ).

% Diff_Int2
tff(fact_3411_Diff__Int2,axiom,
    ! [A2: set_nat,C2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),B2) ).

% Diff_Int2
tff(fact_3412_Diff__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ).

% Diff_Diff_Int
tff(fact_3413_Diff__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ).

% Diff_Diff_Int
tff(fact_3414_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),A2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,C2),B2)) ).

% Diff_Int_distrib
tff(fact_3415_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),A2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,C2),B2)) ).

% Diff_Int_distrib
tff(fact_3416_Diff__Int__distrib2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% Diff_Int_distrib2
tff(fact_3417_Diff__Int__distrib2,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Diff_Int_distrib2
tff(fact_3418_disjoint__alt__simp3,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( aa_set3702741264012733747_nat_o(aa_set7675215768510388063_nat_o(ord_le7866589430770878221at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),A2)
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
tff(fact_3419_disjoint__alt__simp3,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( aa_set_nat_o(aa_set7861427066675100071_nat_o(ord_less_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),A2)
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
tff(fact_3420_disjoint__alt__simp3,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ord_less_set_int(minus_minus_set_int(A2,B2),A2)
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp3
tff(fact_3421_disjoint__alt__simp1,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
tff(fact_3422_disjoint__alt__simp1,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
tff(fact_3423_disjoint__alt__simp1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) = A2 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ).

% disjoint_alt_simp1
tff(fact_3424_disjoint__alt__simp2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) != A2 )
    <=> ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
tff(fact_3425_disjoint__alt__simp2,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) != A2 )
    <=> ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
tff(fact_3426_disjoint__alt__simp2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( minus_minus_set_int(A2,B2) != A2 )
    <=> ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) != bot_bot_set_int ) ) ).

% disjoint_alt_simp2
tff(fact_3427_Diff__triv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat )
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_3428_Diff__triv,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat )
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2) = A2 ) ) ).

% Diff_triv
tff(fact_3429_Diff__triv,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int )
     => ( minus_minus_set_int(A2,B2) = A2 ) ) ).

% Diff_triv
tff(fact_3430_Int__Diff__disjoint,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = bot_bo2099793752762293965at_nat ).

% Int_Diff_disjoint
tff(fact_3431_Int__Diff__disjoint,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = bot_bot_set_nat ).

% Int_Diff_disjoint
tff(fact_3432_Int__Diff__disjoint,axiom,
    ! [A2: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)),minus_minus_set_int(A2,B2)) = bot_bot_set_int ).

% Int_Diff_disjoint
tff(fact_3433_Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C2)) ).

% Diff_Un
tff(fact_3434_Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ).

% Diff_Un
tff(fact_3435_Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),C2)) ).

% Diff_Int
tff(fact_3436_Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat,C2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),C2)) ).

% Diff_Int
tff(fact_3437_Int__Diff__Un,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)) = A2 ).

% Int_Diff_Un
tff(fact_3438_Int__Diff__Un,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)) = A2 ).

% Int_Diff_Un
tff(fact_3439_Un__Diff__Int,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(sup_su6327502436637775413at_nat,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(minus_1356011639430497352at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) = A2 ).

% Un_Diff_Int
tff(fact_3440_Un__Diff__Int,axiom,
    ! [A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(sup_sup_set_nat,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(minus_minus_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) = A2 ).

% Un_Diff_Int
tff(fact_3441_minus__assn__def,axiom,
    ! [A: assn,B: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(minus_minus_assn,A),B) = aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,A),aa_assn_assn(uminus_uminus_assn,B)) ).

% minus_assn_def
tff(fact_3442_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_nat(inc(X)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(X)),one_one_nat) ).

% numeral_inc
tff(fact_3443_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_int(inc(X)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(X)),one_one_int) ).

% numeral_inc
tff(fact_3444_numeral__inc,axiom,
    ! [X: num] : numeral_numeral_rat(inc(X)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(X)),one_one_rat) ).

% numeral_inc
tff(fact_3445_numeral__inc,axiom,
    ! [X: num] : numera6620942414471956472nteger(inc(X)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(X)),one_one_Code_integer) ).

% numeral_inc
tff(fact_3446_numeral__inc,axiom,
    ! [X: num] : numera5444537566228673987atural(inc(X)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(X)),one_one_Code_natural) ).

% numeral_inc
tff(fact_3447_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(plus_plus_num(M2,one))) ).

% diff_numeral_special(4)
tff(fact_3448_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(plus_plus_num(M2,one))) ).

% diff_numeral_special(4)
tff(fact_3449_diff__numeral__special_I4_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(plus_plus_num(M2,one))) ).

% diff_numeral_special(4)
tff(fact_3450_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,numeral_numeral_int(N))) = numeral_numeral_int(plus_plus_num(one,N)) ).

% diff_numeral_special(3)
tff(fact_3451_diff__numeral__special_I3_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(N))) = numeral_numeral_rat(plus_plus_num(one,N)) ).

% diff_numeral_special(3)
tff(fact_3452_diff__numeral__special_I3_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = numera6620942414471956472nteger(plus_plus_num(one,N)) ).

% diff_numeral_special(3)
tff(fact_3453_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_3454_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% nth_Cons_pos
tff(fact_3455_one__plus__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),numeral_numeral_nat(N)) = numeral_numeral_nat(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_3456_one__plus__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),numeral_numeral_int(N)) = numeral_numeral_int(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_3457_one__plus__numeral,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),numeral_numeral_rat(N)) = numeral_numeral_rat(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_3458_one__plus__numeral,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),numera6620942414471956472nteger(N)) = numera6620942414471956472nteger(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_3459_one__plus__numeral,axiom,
    ! [N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),numera5444537566228673987atural(N)) = numera5444537566228673987atural(plus_plus_num(one,N)) ).

% one_plus_numeral
tff(fact_3460_numeral__plus__one,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),one_one_nat) = numeral_numeral_nat(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_3461_numeral__plus__one,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),one_one_int) = numeral_numeral_int(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_3462_numeral__plus__one,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),one_one_rat) = numeral_numeral_rat(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_3463_numeral__plus__one,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(N)),one_one_Code_integer) = numera6620942414471956472nteger(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_3464_numeral__plus__one,axiom,
    ! [N: num] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(N)),one_one_Code_natural) = numera5444537566228673987atural(plus_plus_num(N,one)) ).

% numeral_plus_one
tff(fact_3465_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3466_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_rat_o(ord_less_rat(uminus_uminus_rat(numeral_numeral_rat(M2))),uminus_uminus_rat(one_one_rat))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3467_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer))
    <=> ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_3468_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3469_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ~ aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3470_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: 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,numeral_numeral_int(M2)))
    <=> ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_3471_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_3472_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_nat(N) = one_one_nat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3473_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int(N) = one_one_int )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3474_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat(N) = one_one_rat )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3475_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger(N) = one_one_Code_integer )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3476_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( numera5444537566228673987atural(N) = one_one_Code_natural )
    <=> ( N = one ) ) ).

% numeral_eq_one_iff
tff(fact_3477_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat = numeral_numeral_nat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3478_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int = numeral_numeral_int(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3479_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat = numeral_numeral_rat(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3480_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer = numera6620942414471956472nteger(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3481_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_natural = numera5444537566228673987atural(N) )
    <=> ( one = N ) ) ).

% one_eq_numeral_iff
tff(fact_3482_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] : aa_nat_int(nth_int(cons_int(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_3483_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),zero_zero_nat) = X ).

% nth_Cons_0
tff(fact_3484_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] : aa_nat_int(nth_int(cons_int(X,Xs)),aa_nat_nat(suc,N)) = aa_nat_int(nth_int(Xs),N) ).

% nth_Cons_Suc
tff(fact_3485_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),aa_nat_nat(suc,N)) = aa_nat_nat(nth_nat(Xs),N) ).

% nth_Cons_Suc
tff(fact_3486_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_3487_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] : enumerate_int(N,cons_int(X,Xs)) = cons_P2335045147070616083at_int(product_Pair_nat_int(N,X),enumerate_int(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_3488_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] : enumerate_nat(N,cons_nat(X,Xs)) = cons_P6512896166579812791at_nat(product_Pair_nat_nat(N,X),enumerate_nat(aa_nat_nat(suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_3489_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(numera6620942414471956472nteger(N)),one_one_Code_integer)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_3490_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(numera5444537566228673987atural(N)),one_one_Code_natural)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_3491_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(ord_less_eq_rat(numeral_numeral_rat(N)),one_one_rat)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_3492_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(N)),one_one_nat)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_3493_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(N)),one_one_int)
    <=> ord_less_eq_num(N,one) ) ).

% numeral_le_one_iff
tff(fact_3494_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),numeral_numeral_nat(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_3495_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),numeral_numeral_int(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_3496_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_rat_o(ord_less_rat(one_one_rat),numeral_numeral_rat(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_3497_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),numera6620942414471956472nteger(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_3498_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(one_one_Code_natural),numera5444537566228673987atural(N))
    <=> ord_less_num(one,N) ) ).

% one_less_numeral_iff
tff(fact_3499_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3500_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_3501_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( N = one ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3502_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_int_int(uminus_uminus_int,one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3503_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_3504_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N)) )
    <=> ( N = one ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3505_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_diff
tff(fact_3506_nth__Cons__numeral,axiom,
    ! [X: int,Xs: list_int,V2: num] : aa_nat_int(nth_int(cons_int(X,Xs)),numeral_numeral_nat(V2)) = aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V2)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_3507_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V2: num] : aa_nat_nat(nth_nat(cons_nat(X,Xs)),numeral_numeral_nat(V2)) = aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V2)),one_one_nat)) ).

% nth_Cons_numeral
tff(fact_3508_power__Suc,axiom,
    ! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(A),N)) ).

% power_Suc
tff(fact_3509_power__Suc,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)) ).

% power_Suc
tff(fact_3510_power__Suc,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(A),N)) ).

% power_Suc
tff(fact_3511_power__Suc,axiom,
    ! [A: code_integer,N: nat] : power_8256067586552552935nteger(A,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(A,N)) ).

% power_Suc
tff(fact_3512_power__Suc,axiom,
    ! [A: code_natural,N: nat] : power_7079662738309270450atural(A,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(A,N)) ).

% power_Suc
tff(fact_3513_power__Suc2,axiom,
    ! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,N)) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_nat_assn(power_power_assn(A),N)),A) ).

% power_Suc2
tff(fact_3514_power__Suc2,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),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,A),N)),A) ).

% power_Suc2
tff(fact_3515_power__Suc2,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_nat_int(power_power_int(A),N)),A) ).

% power_Suc2
tff(fact_3516_power__Suc2,axiom,
    ! [A: code_integer,N: nat] : power_8256067586552552935nteger(A,aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(A,N)),A) ).

% power_Suc2
tff(fact_3517_power__Suc2,axiom,
    ! [A: code_natural,N: nat] : power_7079662738309270450atural(A,aa_nat_nat(suc,N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,power_7079662738309270450atural(A,N)),A) ).

% power_Suc2
tff(fact_3518_One__nat__def,axiom,
    one_one_nat = aa_nat_nat(suc,zero_zero_nat) ).

% One_nat_def
tff(fact_3519_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_3520_plus__1__eq__Suc,axiom,
    aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat) = suc ).

% plus_1_eq_Suc
tff(fact_3521_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_3522_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),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,M2),one_one_nat)),N) ).

% diff_Suc_eq_diff_pred
tff(fact_3523_mult__numeral__1__right,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(one)) = A ).

% mult_numeral_1_right
tff(fact_3524_mult__numeral__1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(one)) = A ).

% mult_numeral_1_right
tff(fact_3525_mult__numeral__1__right,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),numeral_numeral_rat(one)) = A ).

% mult_numeral_1_right
tff(fact_3526_mult__numeral__1__right,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(one)) = A ).

% mult_numeral_1_right
tff(fact_3527_mult__numeral__1__right,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),numera5444537566228673987atural(one)) = A ).

% mult_numeral_1_right
tff(fact_3528_mult__numeral__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(one)),A) = A ).

% mult_numeral_1
tff(fact_3529_mult__numeral__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(one)),A) = A ).

% mult_numeral_1
tff(fact_3530_mult__numeral__1,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(one)),A) = A ).

% mult_numeral_1
tff(fact_3531_mult__numeral__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(one)),A) = A ).

% mult_numeral_1
tff(fact_3532_mult__numeral__1,axiom,
    ! [A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(one)),A) = A ).

% mult_numeral_1
tff(fact_3533_numeral__One,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numeral_One
tff(fact_3534_numeral__One,axiom,
    numeral_numeral_int(one) = one_one_int ).

% numeral_One
tff(fact_3535_numeral__One,axiom,
    numeral_numeral_rat(one) = one_one_rat ).

% numeral_One
tff(fact_3536_numeral__One,axiom,
    numera6620942414471956472nteger(one) = one_one_Code_integer ).

% numeral_One
tff(fact_3537_numeral__One,axiom,
    numera5444537566228673987atural(one) = one_one_Code_natural ).

% numeral_One
tff(fact_3538_numerals_I1_J,axiom,
    numeral_numeral_nat(one) = one_one_nat ).

% numerals(1)
tff(fact_3539_gen__length__code_I2_J,axiom,
    ! [N: nat,X: int,Xs: list_int] : gen_length_int(N,cons_int(X,Xs)) = gen_length_int(aa_nat_nat(suc,N),Xs) ).

% gen_length_code(2)
tff(fact_3540_gen__length__code_I2_J,axiom,
    ! [N: nat,X: nat,Xs: list_nat] : gen_length_nat(N,cons_nat(X,Xs)) = gen_length_nat(aa_nat_nat(suc,N),Xs) ).

% gen_length_code(2)
tff(fact_3541_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),A)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(one_one_Code_integer),power_8256067586552552935nteger(A,aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3542_power__gt1,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(one_one_rat),A)
     => aa_rat_o(ord_less_rat(one_one_rat),power_power_rat(A,aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3543_power__gt1,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),A)
     => 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,A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3544_power__gt1,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,one_one_int),aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))) ) ).

% power_gt1
tff(fact_3545_nat__induct__non__zero,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N3)
             => ( aa_nat_o(Pa,N3)
               => aa_nat_o(Pa,aa_nat_nat(suc,N3)) ) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_3546_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(one))),B) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(1)
tff(fact_3547_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,uminus_uminus_rat(numeral_numeral_rat(one))),B) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(1)
tff(fact_3548_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(1)
tff(fact_3549_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(uminus_uminus_int,numeral_numeral_int(one))) = aa_int_int(uminus_uminus_int,B) ).

% mult_1s_ring_1(2)
tff(fact_3550_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),uminus_uminus_rat(numeral_numeral_rat(one))) = uminus_uminus_rat(B) ).

% mult_1s_ring_1(2)
tff(fact_3551_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,B) ).

% mult_1s_ring_1(2)
tff(fact_3552_uminus__numeral__One,axiom,
    aa_int_int(uminus_uminus_int,numeral_numeral_int(one)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% uminus_numeral_One
tff(fact_3553_uminus__numeral__One,axiom,
    uminus_uminus_rat(numeral_numeral_rat(one)) = uminus_uminus_rat(one_one_rat) ).

% uminus_numeral_One
tff(fact_3554_uminus__numeral__One,axiom,
    aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(one)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% uminus_numeral_One
tff(fact_3555_nth__Cons_H,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      aa_nat_int(nth_int(cons_int(X,Xs)),N) = $ite(N = zero_zero_nat,X,aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% nth_Cons'
tff(fact_3556_nth__Cons_H,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = $ite(N = zero_zero_nat,X,aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% nth_Cons'
tff(fact_3557_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),one_one_Code_integer)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(A,aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3558_power__Suc__le__self,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le1926595141338095240atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le1926595141338095240atural(A),one_one_Code_natural)
       => aa_Code_natural_o(ord_le1926595141338095240atural(power_7079662738309270450atural(A,aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3559_power__Suc__le__self,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_eq_rat(A),one_one_rat)
       => aa_rat_o(ord_less_eq_rat(power_power_rat(A,aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3560_power__Suc__le__self,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),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,A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3561_power__Suc__le__self,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),A) ) ) ).

% power_Suc_le_self
tff(fact_3562_power__Suc__less__one,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(ord_le5570908160329646204atural(zero_z2226904508553997617atural),A)
     => ( aa_Code_natural_o(ord_le5570908160329646204atural(A),one_one_Code_natural)
       => aa_Code_natural_o(ord_le5570908160329646204atural(power_7079662738309270450atural(A,aa_nat_nat(suc,N))),one_one_Code_natural) ) ) ).

% power_Suc_less_one
tff(fact_3563_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),one_one_Code_integer)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(A,aa_nat_nat(suc,N))),one_one_Code_integer) ) ) ).

% power_Suc_less_one
tff(fact_3564_power__Suc__less__one,axiom,
    ! [A: rat,N: nat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( aa_rat_o(ord_less_rat(A),one_one_rat)
       => aa_rat_o(ord_less_rat(power_power_rat(A,aa_nat_nat(suc,N))),one_one_rat) ) ) ).

% power_Suc_less_one
tff(fact_3565_power__Suc__less__one,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),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,A),aa_nat_nat(suc,N))),one_one_nat) ) ) ).

% power_Suc_less_one
tff(fact_3566_power__Suc__less__one,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(A),aa_nat_nat(suc,N))),one_one_int) ) ) ).

% power_Suc_less_one
tff(fact_3567_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: 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,M2)),N) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_3568_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_3569_add__eq__if,axiom,
    ! [M2: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N) = $ite(M2 = 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,M2),one_one_nat)),N))) ).

% add_eq_if
tff(fact_3570_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),M2)
       => ( aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_3571_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_int(nth_int(cons_int(X,Xs)),N) = Y )
      <=> ( ( aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3572_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( aa_nat_nat(nth_nat(cons_nat(X,Xs)),N) = Y )
      <=> ( ( aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) = Y )
          & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_3573_one__div__numeral,axiom,
    ! [N: num] : divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(N)) = produc8508995932063986495nteger(unique3479559517661332726nteger(one,N)) ).

% one_div_numeral
tff(fact_3574_one__div__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(N)) = aa_Pro4243173770044397973at_nat(product_fst_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_div_numeral
tff(fact_3575_one__div__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(N)) = aa_Pro6799258434465214121nt_int(product_fst_int_int,unique5052692396658037445od_int(one,N)) ).

% one_div_numeral
tff(fact_3576_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,numeral_numeral_nat(bit0(one))),N)) = one_one_int ).

% power_minus1_even
tff(fact_3577_power__minus1__even,axiom,
    ! [N: nat] : power_power_rat(uminus_uminus_rat(one_one_rat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) = one_one_rat ).

% power_minus1_even
tff(fact_3578_power__minus1__even,axiom,
    ! [N: nat] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) = one_one_Code_integer ).

% power_minus1_even
tff(fact_3579_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,numeral_numeral_nat(bit0(one))),N))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% power_minus1_odd
tff(fact_3580_power__minus1__odd,axiom,
    ! [N: nat] : power_power_rat(uminus_uminus_rat(one_one_rat),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = uminus_uminus_rat(one_one_rat) ).

% power_minus1_odd
tff(fact_3581_power__minus1__odd,axiom,
    ! [N: nat] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% power_minus1_odd
tff(fact_3582_power2__diff,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Y)),numeral_numeral_nat(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(X),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3583_power2__diff,axiom,
    ! [X: rat,Y: rat] : power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X),Y),numeral_numeral_nat(bit0(one))) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,power_power_rat(X,numeral_numeral_nat(bit0(one)))),power_power_rat(Y,numeral_numeral_nat(bit0(one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3584_power2__diff,axiom,
    ! [X: code_integer,Y: code_integer] : power_8256067586552552935nteger(minus_8373710615458151222nteger(X,Y),numeral_numeral_nat(bit0(one))) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one)))),power_8256067586552552935nteger(Y,numeral_numeral_nat(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),X)),Y)) ).

% power2_diff
tff(fact_3585_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(one,M2) ).

% diff_numeral_special(8)
tff(fact_3586_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),uminus_uminus_rat(one_one_rat)) = neg_numeral_sub_rat(one,M2) ).

% diff_numeral_special(8)
tff(fact_3587_diff__numeral__special_I8_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(one,M2) ).

% diff_numeral_special(8)
tff(fact_3588_diff__numeral__special_I7_J,axiom,
    ! [N: num] : 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,numeral_numeral_int(N))) = neg_numeral_sub_int(N,one) ).

% diff_numeral_special(7)
tff(fact_3589_diff__numeral__special_I7_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(numeral_numeral_rat(N))) = neg_numeral_sub_rat(N,one) ).

% diff_numeral_special(7)
tff(fact_3590_diff__numeral__special_I7_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(N))) = neg_nu5755505904847501662nteger(N,one) ).

% diff_numeral_special(7)
tff(fact_3591_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,neg_numeral_sub_int(M2,one))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)) ).

% minus_sub_one_diff_one
tff(fact_3592_minus__sub__one__diff__one,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(neg_numeral_sub_rat(M2,one))),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(M2)) ).

% minus_sub_one_diff_one
tff(fact_3593_minus__sub__one__diff__one,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,neg_nu5755505904847501662nteger(M2,one)),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2)) ).

% minus_sub_one_diff_one
tff(fact_3594_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))) = neg_numeral_sub_int(one,M2) ).

% add_neg_numeral_special(1)
tff(fact_3595_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),uminus_uminus_rat(numeral_numeral_rat(M2))) = neg_numeral_sub_rat(one,M2) ).

% add_neg_numeral_special(1)
tff(fact_3596_add__neg__numeral__special_I1_J,axiom,
    ! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))) = neg_nu5755505904847501662nteger(one,M2) ).

% add_neg_numeral_special(1)
tff(fact_3597_one__add__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% one_add_one
tff(fact_3598_one__add__one,axiom,
    aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),one_one_int) = numeral_numeral_int(bit0(one)) ).

% one_add_one
tff(fact_3599_one__add__one,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% one_add_one
tff(fact_3600_one__add__one,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),one_one_Code_integer) = numera6620942414471956472nteger(bit0(one)) ).

% one_add_one
tff(fact_3601_one__add__one,axiom,
    aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),one_one_Code_natural) = numera5444537566228673987atural(bit0(one)) ).

% one_add_one
tff(fact_3602_Suc__1,axiom,
    aa_nat_nat(suc,one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% Suc_1
tff(fact_3603_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),one_one_int) = neg_numeral_sub_int(M2,one) ).

% diff_numeral_special(2)
tff(fact_3604_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(M2)),one_one_rat) = neg_numeral_sub_rat(M2,one) ).

% diff_numeral_special(2)
tff(fact_3605_diff__numeral__special_I2_J,axiom,
    ! [M2: num] : minus_8373710615458151222nteger(numera6620942414471956472nteger(M2),one_one_Code_integer) = neg_nu5755505904847501662nteger(M2,one) ).

% diff_numeral_special(2)
tff(fact_3606_diff__numeral__special_I1_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),numeral_numeral_int(N)) = neg_numeral_sub_int(one,N) ).

% diff_numeral_special(1)
tff(fact_3607_diff__numeral__special_I1_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),numeral_numeral_rat(N)) = neg_numeral_sub_rat(one,N) ).

% diff_numeral_special(1)
tff(fact_3608_diff__numeral__special_I1_J,axiom,
    ! [N: num] : minus_8373710615458151222nteger(one_one_Code_integer,numera6620942414471956472nteger(N)) = neg_nu5755505904847501662nteger(one,N) ).

% diff_numeral_special(1)
tff(fact_3609_one__div__two__eq__zero,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% one_div_two_eq_zero
tff(fact_3610_one__div__two__eq__zero,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% one_div_two_eq_zero
tff(fact_3611_one__div__two__eq__zero,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% one_div_two_eq_zero
tff(fact_3612_one__div__two__eq__zero,axiom,
    divide5121882707175180666atural(one_one_Code_natural,numera5444537566228673987atural(bit0(one))) = zero_z2226904508553997617atural ).

% one_div_two_eq_zero
tff(fact_3613_bits__1__div__2,axiom,
    divide6298287555418463151nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ).

% bits_1_div_2
tff(fact_3614_bits__1__div__2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = zero_zero_nat ).

% bits_1_div_2
tff(fact_3615_bits__1__div__2,axiom,
    aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),numeral_numeral_int(bit0(one))) = zero_zero_int ).

% bits_1_div_2
tff(fact_3616_bits__1__div__2,axiom,
    divide5121882707175180666atural(one_one_Code_natural,numera5444537566228673987atural(bit0(one))) = zero_z2226904508553997617atural ).

% bits_1_div_2
tff(fact_3617_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,numeral_numeral_int(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3618_add__neg__numeral__special_I9_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3619_add__neg__numeral__special_I9_J,axiom,
    aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(bit0(one))) ).

% add_neg_numeral_special(9)
tff(fact_3620_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,numeral_numeral_int(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3621_diff__numeral__special_I10_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,uminus_uminus_rat(one_one_rat)),one_one_rat) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3622_diff__numeral__special_I10_J,axiom,
    minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(bit0(one))) ).

% diff_numeral_special(10)
tff(fact_3623_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)) = numeral_numeral_int(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3624_diff__numeral__special_I11_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),uminus_uminus_rat(one_one_rat)) = numeral_numeral_rat(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3625_diff__numeral__special_I11_J,axiom,
    minus_8373710615458151222nteger(one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = numera6620942414471956472nteger(bit0(one)) ).

% diff_numeral_special(11)
tff(fact_3626_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)),numeral_numeral_int(bit0(one))) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_2_eq
tff(fact_3627_minus__1__div__2__eq,axiom,
    divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% minus_1_div_2_eq
tff(fact_3628_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,one_one_int)),numeral_numeral_int(N)) = neg_numeral_sub_int(N,one) ).

% add_neg_numeral_special(4)
tff(fact_3629_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(one_one_rat)),numeral_numeral_rat(N)) = neg_numeral_sub_rat(N,one) ).

% add_neg_numeral_special(4)
tff(fact_3630_add__neg__numeral__special_I4_J,axiom,
    ! [N: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),numera6620942414471956472nteger(N)) = neg_nu5755505904847501662nteger(N,one) ).

% add_neg_numeral_special(4)
tff(fact_3631_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,one_one_int)) = neg_numeral_sub_int(M2,one) ).

% add_neg_numeral_special(3)
tff(fact_3632_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(M2)),uminus_uminus_rat(one_one_rat)) = neg_numeral_sub_rat(M2,one) ).

% add_neg_numeral_special(3)
tff(fact_3633_add__neg__numeral__special_I3_J,axiom,
    ! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(M2)),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = neg_nu5755505904847501662nteger(M2,one) ).

% add_neg_numeral_special(3)
tff(fact_3634_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int) = neg_numeral_sub_int(one,M2) ).

% add_neg_numeral_special(2)
tff(fact_3635_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,uminus_uminus_rat(numeral_numeral_rat(M2))),one_one_rat) = neg_numeral_sub_rat(one,M2) ).

% add_neg_numeral_special(2)
tff(fact_3636_add__neg__numeral__special_I2_J,axiom,
    ! [M2: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(M2))),one_one_Code_integer) = neg_nu5755505904847501662nteger(one,M2) ).

% add_neg_numeral_special(2)
tff(fact_3637_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( 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,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% pos_zdiv_mult_2
tff(fact_3638_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),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,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A) ) ) ).

% neg_zdiv_mult_2
tff(fact_3639_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(numeral_numeral_int(bit0(one))),N)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% minus_1_div_exp_eq_int
tff(fact_3640_int__bit__induct,axiom,
    ! [Pa: fun_int_o,K: int] :
      ( aa_int_o(Pa,zero_zero_int)
     => ( aa_int_o(Pa,aa_int_int(uminus_uminus_int,one_one_int))
       => ( ! [K2: int] :
              ( aa_int_o(Pa,K2)
             => ( ( K2 != zero_zero_int )
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(times_times_int,K2),numeral_numeral_int(bit0(one)))) ) )
         => ( ! [K2: int] :
                ( aa_int_o(Pa,K2)
               => ( ( K2 != aa_int_int(uminus_uminus_int,one_one_int) )
                 => aa_int_o(Pa,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),numeral_numeral_int(bit0(one))))) ) )
           => aa_int_o(Pa,K) ) ) ) ) ).

% int_bit_induct
tff(fact_3641_left__add__twice,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_3642_left__add__twice,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),B)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_3643_left__add__twice,axiom,
    ! [A: rat,B: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_3644_left__add__twice,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_3645_left__add__twice,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),A)),B) ).

% left_add_twice
tff(fact_3646_mult__2__right,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,Z),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2_right
tff(fact_3647_mult__2__right,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,Z),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2_right
tff(fact_3648_mult__2__right,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Z),numeral_numeral_rat(bit0(one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2_right
tff(fact_3649_mult__2__right,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Z),numera6620942414471956472nteger(bit0(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2_right
tff(fact_3650_mult__2__right,axiom,
    ! [Z: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Z),numera5444537566228673987atural(bit0(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z),Z) ).

% mult_2_right
tff(fact_3651_mult__2,axiom,
    ! [Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Z) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Z),Z) ).

% mult_2
tff(fact_3652_mult__2,axiom,
    ! [Z: int] : aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Z) = aa_int_int(aa_int_fun_int_int(plus_plus_int,Z),Z) ).

% mult_2
tff(fact_3653_mult__2,axiom,
    ! [Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(bit0(one))),Z) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Z),Z) ).

% mult_2
tff(fact_3654_mult__2,axiom,
    ! [Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Z) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Z),Z) ).

% mult_2
tff(fact_3655_mult__2,axiom,
    ! [Z: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),Z) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Z),Z) ).

% mult_2
tff(fact_3656_power2__eq__square,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(bit0(one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A) ).

% power2_eq_square
tff(fact_3657_power2__eq__square,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A) ).

% power2_eq_square
tff(fact_3658_power2__eq__square,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% power2_eq_square
tff(fact_3659_power2__eq__square,axiom,
    ! [A: code_integer] : power_8256067586552552935nteger(A,numeral_numeral_nat(bit0(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% power2_eq_square
tff(fact_3660_power2__eq__square,axiom,
    ! [A: code_natural] : power_7079662738309270450atural(A,numeral_numeral_nat(bit0(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),A) ).

% power2_eq_square
tff(fact_3661_power4__eq__xxxx,axiom,
    ! [X: assn] : aa_nat_assn(power_power_assn(X),numeral_numeral_nat(bit0(bit0(one)))) = 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,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3662_power4__eq__xxxx,axiom,
    ! [X: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),numeral_numeral_nat(bit0(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,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3663_power4__eq__xxxx,axiom,
    ! [X: int] : aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(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,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3664_power4__eq__xxxx,axiom,
    ! [X: code_integer] : power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(bit0(one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3665_power4__eq__xxxx,axiom,
    ! [X: code_natural] : power_7079662738309270450atural(X,numeral_numeral_nat(bit0(bit0(one)))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,X),X)),X)),X) ).

% power4_eq_xxxx
tff(fact_3666_one__power2,axiom,
    power_8256067586552552935nteger(one_one_Code_integer,numeral_numeral_nat(bit0(one))) = one_one_Code_integer ).

% one_power2
tff(fact_3667_one__power2,axiom,
    power_power_rat(one_one_rat,numeral_numeral_nat(bit0(one))) = one_one_rat ).

% one_power2
tff(fact_3668_one__power2,axiom,
    aa_nat_int(power_power_int(one_one_int),numeral_numeral_nat(bit0(one))) = one_one_int ).

% one_power2
tff(fact_3669_one__power2,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% one_power2
tff(fact_3670_nat__1__add__1,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,one_one_nat),one_one_nat) = numeral_numeral_nat(bit0(one)) ).

% nat_1_add_1
tff(fact_3671_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( aa_nat_int(power_power_int(A),numeral_numeral_nat(bit0(one))) = one_one_int )
    <=> ( ( A = one_one_int )
        | ( A = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% power2_eq_1_iff
tff(fact_3672_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( power_power_rat(A,numeral_numeral_nat(bit0(one))) = one_one_rat )
    <=> ( ( A = one_one_rat )
        | ( A = uminus_uminus_rat(one_one_rat) ) ) ) ).

% power2_eq_1_iff
tff(fact_3673_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger(A,numeral_numeral_nat(bit0(one))) = one_one_Code_integer )
    <=> ( ( A = one_one_Code_integer )
        | ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% power2_eq_1_iff
tff(fact_3674_nat__induct2,axiom,
    ! [Pa: fun_nat_o,N: nat] :
      ( aa_nat_o(Pa,zero_zero_nat)
     => ( aa_nat_o(Pa,one_one_nat)
       => ( ! [N3: nat] :
              ( aa_nat_o(Pa,N3)
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),numeral_numeral_nat(bit0(one)))) )
         => aa_nat_o(Pa,N) ) ) ) ).

% nat_induct2
tff(fact_3675_power2__sum,axiom,
    ! [X: 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,X),Y)),numeral_numeral_nat(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,X),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,Y),numeral_numeral_nat(bit0(one))))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_3676_power2__sum,axiom,
    ! [X: int,Y: int] : aa_nat_int(power_power_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),numeral_numeral_nat(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(X),numeral_numeral_nat(bit0(one)))),aa_nat_int(power_power_int(Y),numeral_numeral_nat(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_3677_power2__sum,axiom,
    ! [X: rat,Y: rat] : power_power_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y),numeral_numeral_nat(bit0(one))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,power_power_rat(X,numeral_numeral_nat(bit0(one)))),power_power_rat(Y,numeral_numeral_nat(bit0(one))))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,numeral_numeral_rat(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_3678_power2__sum,axiom,
    ! [X: code_integer,Y: code_integer] : power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y),numeral_numeral_nat(bit0(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one)))),power_8256067586552552935nteger(Y,numeral_numeral_nat(bit0(one))))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_3679_power2__sum,axiom,
    ! [X: code_natural,Y: code_natural] : power_7079662738309270450atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X),Y),numeral_numeral_nat(bit0(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,power_7079662738309270450atural(X,numeral_numeral_nat(bit0(one)))),power_7079662738309270450atural(Y,numeral_numeral_nat(bit0(one))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),X)),Y)) ).

% power2_sum
tff(fact_3680_square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(X),one_one_Code_integer)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one)))),one_one_Code_integer) ) ) ).

% square_le_1
tff(fact_3681_square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(one_one_rat)),X)
     => ( aa_rat_o(ord_less_eq_rat(X),one_one_rat)
       => aa_rat_o(ord_less_eq_rat(power_power_rat(X,numeral_numeral_nat(bit0(one)))),one_one_rat) ) ) ).

% square_le_1
tff(fact_3682_square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,one_one_int)),X)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),one_one_int)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(one)))),one_one_int) ) ) ).

% square_le_1
tff(fact_3683_minus__power__mult__self,axiom,
    ! [A: 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,A)),N)),aa_nat_int(power_power_int(aa_int_int(uminus_uminus_int,A)),N)) = aa_nat_int(power_power_int(A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_3684_minus__power__mult__self,axiom,
    ! [A: rat,N: nat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,power_power_rat(uminus_uminus_rat(A),N)),power_power_rat(uminus_uminus_rat(A),N)) = power_power_rat(A,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_3685_minus__power__mult__self,axiom,
    ! [A: code_integer,N: nat] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),N)),power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),N)) = power_8256067586552552935nteger(A,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N)) ).

% minus_power_mult_self
tff(fact_3686_power__odd__eq,axiom,
    ! [A: assn,N: nat] : aa_nat_assn(power_power_assn(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),aa_nat_assn(power_power_assn(aa_nat_assn(power_power_assn(A),N)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_3687_power__odd__eq,axiom,
    ! [A: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),N)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_3688_power__odd__eq,axiom,
    ! [A: int,N: nat] : aa_nat_int(power_power_int(A),aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(aa_nat_int(power_power_int(A),N)),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_3689_power__odd__eq,axiom,
    ! [A: code_integer,N: nat] : power_8256067586552552935nteger(A,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(power_8256067586552552935nteger(A,N),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_3690_power__odd__eq,axiom,
    ! [A: code_natural,N: nat] : power_7079662738309270450atural(A,aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),N))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(power_7079662738309270450atural(A,N),numeral_numeral_nat(bit0(one)))) ).

% power_odd_eq
tff(fact_3691_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(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,B),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl2
tff(fact_3692_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),B)
     => ( 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,B),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,B),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N3),one_one_nat))) ) ) ) ).

% ex_power_ivl1
tff(fact_3693_divmod__step__eq,axiom,
    ! [L: num,Q2: nat,R2: nat] :
      unique5026877609467782581ep_nat(L,product_Pair_nat_nat(Q2,R2)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(L)),R2),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,numeral_numeral_nat(bit0(one))),Q2)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,R2),numeral_numeral_nat(L))),product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3694_divmod__step__eq,axiom,
    ! [L: num,Q2: int,R2: int] :
      unique5024387138958732305ep_int(L,product_Pair_int_int(Q2,R2)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(L)),R2),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,numeral_numeral_int(bit0(one))),Q2)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,R2),numeral_numeral_int(L))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3695_divmod__step__eq,axiom,
    ! [L: num,Q2: code_integer,R2: code_integer] :
      unique4921790084139445826nteger(L,produc1086072967326762835nteger(Q2,R2)) = $ite(aa_Code_integer_o(ord_le3102999989581377725nteger(numera6620942414471956472nteger(L)),R2),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Q2)),one_one_Code_integer),minus_8373710615458151222nteger(R2,numera6620942414471956472nteger(L))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Q2),R2)) ).

% divmod_step_eq
tff(fact_3696_set__bit__0,axiom,
    ! [A: code_integer] : bit_se2793503036327961859nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% set_bit_0
tff(fact_3697_set__bit__0,axiom,
    ! [A: code_natural] : bit_se1617098188084679374atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))) ).

% set_bit_0
tff(fact_3698_set__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,zero_zero_nat),A) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))) ).

% set_bit_0
tff(fact_3699_set__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se7879613467334960850it_int(zero_zero_nat),A) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))))) ).

% set_bit_0
tff(fact_3700_unset__bit__0,axiom,
    ! [A: code_integer] : bit_se8260200283734997820nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% unset_bit_0
tff(fact_3701_unset__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))) ).

% unset_bit_0
tff(fact_3702_unset__bit__0,axiom,
    ! [A: code_natural] : bit_se7083795435491715335atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))) ).

% unset_bit_0
tff(fact_3703_unset__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se4203085406695923979it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))) ).

% unset_bit_0
tff(fact_3704_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,numeral_numeral_int(bit0(one))) ).

% dbl_simps(4)
tff(fact_3705_dbl__simps_I4_J,axiom,
    neg_numeral_dbl_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(bit0(one))) ).

% dbl_simps(4)
tff(fact_3706_dbl__simps_I4_J,axiom,
    neg_nu8804712462038260780nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(bit0(one))) ).

% dbl_simps(4)
tff(fact_3707_dbl__simps_I3_J,axiom,
    aa_int_int(neg_numeral_dbl_int,one_one_int) = numeral_numeral_int(bit0(one)) ).

% dbl_simps(3)
tff(fact_3708_dbl__simps_I3_J,axiom,
    neg_numeral_dbl_rat(one_one_rat) = numeral_numeral_rat(bit0(one)) ).

% dbl_simps(3)
tff(fact_3709_dbl__simps_I3_J,axiom,
    neg_nu8804712462038260780nteger(one_one_Code_integer) = numera6620942414471956472nteger(bit0(one)) ).

% dbl_simps(3)
tff(fact_3710_even__succ__div__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3711_even__succ__div__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3712_even__succ__div__exp,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( 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),A)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3713_even__succ__div__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = divide5121882707175180666atural(A,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) ) ) ) ).

% even_succ_div_exp
tff(fact_3714_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2)),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
        | ( power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N) = zero_z3403309356797280102nteger )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
          & aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3715_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
        | ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N) = zero_zero_nat )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
          & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3716_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
        | ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) = zero_zero_int )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
          & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3717_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_natural,M2: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2)),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2)
        | ( power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N) = zero_z2226904508553997617atural )
        | ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
          & aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_3718_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)))
         => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)))),one_one_Code_integer) = divide6298287555418463151nteger(A,B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3719_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)))),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3720_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)))),one_one_int) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_3721_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),one_one_nat)
    <=> ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
tff(fact_3722_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != 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,B),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3723_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != 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,B),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),A))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3724_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A))
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3725_dvd__times__right__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(B),C) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_3726_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != 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,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3727_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != 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,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,A),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3728_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C))
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3729_dvd__times__left__cancel__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(B),C) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_3730_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),C)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,B),C))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(dvd_dvd_rat(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3731_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3732_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(dvd_dvd_Code_integer(A),B) ) ) ).

% dvd_mult_cancel_right
tff(fact_3733_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(dvd_dvd_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),B))
    <=> ( ( C = zero_zero_rat )
        | aa_rat_o(dvd_dvd_rat(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3734_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
    <=> ( ( C = zero_zero_int )
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3735_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
    <=> ( ( C = zero_z3403309356797280102nteger )
        | aa_Code_integer_o(dvd_dvd_Code_integer(A),B) ) ) ).

% dvd_mult_cancel_left
tff(fact_3736_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,B),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)))
    <=> aa_rat_o(dvd_dvd_rat(A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3737_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3738_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,B),aa_int_int(aa_int_fun_int_int(times_times_int,C),A)))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3739_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)))
    <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3740_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,B),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)))
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),B) ) ).

% dvd_add_times_triv_right_iff
tff(fact_3741_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( aa_rat_o(dvd_dvd_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),A)),B))
    <=> aa_rat_o(dvd_dvd_rat(A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3742_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),B))
    <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3743_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),B))
    <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3744_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A)),B))
    <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3745_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A)),B))
    <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),B) ) ).

% dvd_add_times_triv_left_iff
tff(fact_3746_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),B)),one_one_nat) ) ) ).

% unit_prod
tff(fact_3747_unit__prod,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),B)),one_one_int) ) ) ).

% unit_prod
tff(fact_3748_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
       => aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer) ) ) ).

% unit_prod
tff(fact_3749_unit__prod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),one_one_Code_natural) ) ) ).

% unit_prod
tff(fact_3750_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),B) = zero_z3403309356797280102nteger ).

% mod_mult_self2_is_0
tff(fact_3751_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),B) = zero_zero_nat ).

% mod_mult_self2_is_0
tff(fact_3752_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),B) = zero_zero_int ).

% mod_mult_self2_is_0
tff(fact_3753_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),B) = zero_z2226904508553997617atural ).

% mod_mult_self2_is_0
tff(fact_3754_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),B) = zero_z3403309356797280102nteger ).

% mod_mult_self1_is_0
tff(fact_3755_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),B) = zero_zero_nat ).

% mod_mult_self1_is_0
tff(fact_3756_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),B) = zero_zero_int ).

% mod_mult_self1_is_0
tff(fact_3757_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A),B) = zero_z2226904508553997617atural ).

% mod_mult_self1_is_0
tff(fact_3758_mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% mod_by_1
tff(fact_3759_mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% mod_by_1
tff(fact_3760_mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% mod_by_1
tff(fact_3761_mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% mod_by_1
tff(fact_3762_bits__mod__by__1,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,one_one_Code_integer) = zero_z3403309356797280102nteger ).

% bits_mod_by_1
tff(fact_3763_bits__mod__by__1,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),one_one_nat) = zero_zero_nat ).

% bits_mod_by_1
tff(fact_3764_bits__mod__by__1,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),one_one_int) = zero_zero_int ).

% bits_mod_by_1
tff(fact_3765_bits__mod__by__1,axiom,
    ! [A: code_natural] : modulo8411746178871703098atural(A,one_one_Code_natural) = zero_z2226904508553997617atural ).

% bits_mod_by_1
tff(fact_3766_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self4
tff(fact_3767_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: 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,B),C)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self4
tff(fact_3768_mod__mult__self4,axiom,
    ! [B: int,C: int,A: 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,B),C)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self4
tff(fact_3769_mod__mult__self4,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self4
tff(fact_3770_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),A),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self3
tff(fact_3771_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: 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),B)),A)),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self3
tff(fact_3772_mod__mult__self3,axiom,
    ! [C: int,B: int,A: 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),B)),A)),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self3
tff(fact_3773_mod__mult__self3,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),A),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self3
tff(fact_3774_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self2
tff(fact_3775_mod__mult__self2,axiom,
    ! [A: nat,B: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self2
tff(fact_3776_mod__mult__self2,axiom,
    ! [A: int,B: 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self2
tff(fact_3777_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self2
tff(fact_3778_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)),B) = modulo364778990260209775nteger(A,B) ).

% mod_mult_self1
tff(fact_3779_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% mod_mult_self1
tff(fact_3780_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% mod_mult_self1
tff(fact_3781_mod__mult__self1,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)),B) = modulo8411746178871703098atural(A,B) ).

% mod_mult_self1
tff(fact_3782_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3783_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3784_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3785_dvd__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(B,A)),A) = B ) ) ).

% dvd_div_mult_self
tff(fact_3786_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3787_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3788_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3789_dvd__mult__div__cancel,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),divide5121882707175180666atural(B,A)) = B ) ) ).

% dvd_mult_div_cancel
tff(fact_3790_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
       => aa_Code_integer_o(dvd_dvd_Code_integer(divide6298287555418463151nteger(A,B)),one_one_Code_integer) ) ) ).

% unit_div
tff(fact_3791_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),B)),one_one_nat) ) ) ).

% unit_div
tff(fact_3792_unit__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),B)),one_one_int) ) ) ).

% unit_div
tff(fact_3793_unit__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),one_one_Code_natural) ) ) ).

% unit_div
tff(fact_3794_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => aa_Code_integer_o(dvd_dvd_Code_integer(divide6298287555418463151nteger(one_one_Code_integer,A)),one_one_Code_integer) ) ).

% unit_div_1_unit
tff(fact_3795_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),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),A)),one_one_nat) ) ).

% unit_div_1_unit
tff(fact_3796_unit__div__1__unit,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),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),A)),one_one_int) ) ).

% unit_div_1_unit
tff(fact_3797_unit__div__1__unit,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(one_one_Code_natural,A)),one_one_Code_natural) ) ).

% unit_div_1_unit
tff(fact_3798_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( divide6298287555418463151nteger(one_one_Code_integer,divide6298287555418463151nteger(one_one_Code_integer,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3799_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),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),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3800_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),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),A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3801_unit__div__1__div__1,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( divide5121882707175180666atural(one_one_Code_natural,divide5121882707175180666atural(one_one_Code_natural,A)) = A ) ) ).

% unit_div_1_div_1
tff(fact_3802_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3803_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),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,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3804_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),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,B),A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3805_unit__div__mult__self,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(B,A)),A) = B ) ) ).

% unit_div_mult_self
tff(fact_3806_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(one_one_Code_integer,A)) = divide6298287555418463151nteger(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3807_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3808_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) ) ) ).

% unit_mult_div_div
tff(fact_3809_unit__mult__div__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(one_one_Code_natural,A)) = divide5121882707175180666atural(B,A) ) ) ).

% unit_mult_div_div
tff(fact_3810_mod__minus1__right,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% mod_minus1_right
tff(fact_3811_mod__minus1__right,axiom,
    ! [A: code_integer] : modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% mod_minus1_right
tff(fact_3812_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: 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,numeral_numeral_nat(K)),N))),numeral_numeral_nat(K)) = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_3813_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
        | aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_3814_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(times_times_int,A),B))
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
        | aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_3815_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B))
    <=> ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
        | aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_3816_even__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B))
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
        | aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),B) ) ) ).

% even_mult_iff
tff(fact_3817_bits__one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% bits_one_mod_two_eq_one
tff(fact_3818_bits__one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% bits_one_mod_two_eq_one
tff(fact_3819_bits__one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(bit0(one))) = one_one_int ).

% bits_one_mod_two_eq_one
tff(fact_3820_bits__one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(bit0(one))) = one_one_Code_natural ).

% bits_one_mod_two_eq_one
tff(fact_3821_one__mod__two__eq__one,axiom,
    modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% one_mod_two_eq_one
tff(fact_3822_one__mod__two__eq__one,axiom,
    aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(bit0(one))) = one_one_nat ).

% one_mod_two_eq_one
tff(fact_3823_one__mod__two__eq__one,axiom,
    aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(bit0(one))) = one_one_int ).

% one_mod_two_eq_one
tff(fact_3824_one__mod__two__eq__one,axiom,
    modulo8411746178871703098atural(one_one_Code_natural,numera5444537566228673987atural(bit0(one))) = one_one_Code_natural ).

% one_mod_two_eq_one
tff(fact_3825_even__plus__one__iff,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3826_even__plus__one__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int))
    <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3827_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer))
    <=> ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3828_even__plus__one__iff,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural))
    <=> ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A) ) ).

% even_plus_one_iff
tff(fact_3829_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3830_not__mod__2__eq__0__eq__1,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != zero_zero_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3831_not__mod__2__eq__0__eq__1,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) != zero_zero_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) = one_one_int ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3832_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) != zero_z2226904508553997617atural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_3833_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != one_one_Code_integer )
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3834_not__mod__2__eq__1__eq__0,axiom,
    ! [A: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != one_one_nat )
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = zero_zero_nat ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3835_not__mod__2__eq__1__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) != one_one_int )
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) = zero_zero_int ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3836_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) != one_one_Code_natural )
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_3837_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)),numeral_numeral_int(bit0(one))) = one_one_int ).

% bits_minus_1_mod_2_eq
tff(fact_3838_bits__minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% bits_minus_1_mod_2_eq
tff(fact_3839_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)),numeral_numeral_int(bit0(one))) = one_one_int ).

% minus_1_mod_2_eq
tff(fact_3840_minus__1__mod__2__eq,axiom,
    modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ).

% minus_1_mod_2_eq
tff(fact_3841_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3842_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
     => ( power_power_rat(uminus_uminus_rat(one_one_rat),N) = one_one_rat ) ) ).

% neg_one_even_power
tff(fact_3843_neg__one__even__power,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N) = one_one_Code_integer ) ) ).

% neg_one_even_power
tff(fact_3844_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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_3845_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
     => ( power_power_rat(uminus_uminus_rat(one_one_rat),N) = uminus_uminus_rat(one_one_rat) ) ) ).

% neg_one_odd_power
tff(fact_3846_neg__one__odd__power,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
     => ( power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ).

% neg_one_odd_power
tff(fact_3847_mod2__gr__0,axiom,
    ! [M2: 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,M2),numeral_numeral_nat(bit0(one))))
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,M2),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% mod2_gr_0
tff(fact_3848_one__mod__numeral,axiom,
    ! [N: num] : modulo364778990260209775nteger(one_one_Code_integer,numera6620942414471956472nteger(N)) = produc6174133586879617921nteger(unique3479559517661332726nteger(one,N)) ).

% one_mod_numeral
tff(fact_3849_one__mod__numeral,axiom,
    ! [N: num] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),numeral_numeral_nat(N)) = aa_Pro4243173770044397973at_nat(product_snd_nat_nat,unique5055182867167087721od_nat(one,N)) ).

% one_mod_numeral
tff(fact_3850_one__mod__numeral,axiom,
    ! [N: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),numeral_numeral_int(N)) = aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)) ).

% one_mod_numeral
tff(fact_3851_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A),numera6620942414471956472nteger(bit0(one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3852_even__succ__div__2,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( 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),A)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3853_even__succ__div__2,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( 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),A)),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3854_even__succ__div__2,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),numera5444537566228673987atural(bit0(one))) = divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))) ) ) ).

% even_succ_div_2
tff(fact_3855_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))),one_one_Code_integer) ) ) ).

% odd_succ_div_two
tff(fact_3856_odd__succ__div__two,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))),one_one_nat) ) ) ).

% odd_succ_div_two
tff(fact_3857_odd__succ__div__two,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))),one_one_int) ) ) ).

% odd_succ_div_two
tff(fact_3858_odd__succ__div__two,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural),numera5444537566228673987atural(bit0(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))),one_one_Code_natural) ) ) ).

% odd_succ_div_two
tff(fact_3859_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),one_one_Code_integer),numera6620942414471956472nteger(bit0(one))) = divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3860_even__succ__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),one_one_nat)),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3861_even__succ__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int)),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3862_even__succ__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),one_one_Code_natural),numera5444537566228673987atural(bit0(one))) = divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))) ) ) ).

% even_succ_div_two
tff(fact_3863_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),numeral_numeral_nat(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_3864_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))),one_one_Code_integer) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3865_odd__two__times__div__two__succ,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))),one_one_nat) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3866_odd__two__times__div__two__succ,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))))),one_one_int) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3867_odd__two__times__div__two__succ,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))),one_one_Code_natural) = A ) ) ).

% odd_two_times_div_two_succ
tff(fact_3868_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)),one_one_nat))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3869_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),one_one_int))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3870_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N),one_one_Code_integer))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3871_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),minus_7197305767214868737atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N),one_one_Code_natural))
    <=> ( N = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_3872_even__succ__mod__exp,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),A),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3873_even__succ__mod__exp,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( 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),A)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3874_even__succ__mod__exp,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( 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),A)),aa_nat_int(power_power_int(numeral_numeral_int(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,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3875_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),A),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N))) ) ) ) ).

% even_succ_mod_exp
tff(fact_3876_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_3877_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_3878_zmod__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(aa_int_fun_int_int(minus_minus_int,B),one_one_int) ) ) ).

% zmod_minus1
tff(fact_3879_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_3880_int__div__less__self,axiom,
    ! [X: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),X)
     => ( 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,X),K)),X) ) ) ).

% int_div_less_self
tff(fact_3881_div__eq__minus1,axiom,
    ! [B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,one_one_int)),B) = aa_int_int(uminus_uminus_int,one_one_int) ) ) ).

% div_eq_minus1
tff(fact_3882_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(uminus_uminus_int,A)),B) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),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,A),B))),one_one_int)) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_3883_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(uminus_uminus_int,B)) = $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),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,A),B))),one_one_int)) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_3884_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( modulo364778990260209775nteger(A,B) = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
tff(fact_3885_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
tff(fact_3886_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
tff(fact_3887_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( modulo8411746178871703098atural(A,B) = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
tff(fact_3888_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_right_eq
tff(fact_3889_mod__mult__right__eq,axiom,
    ! [A: nat,B: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_right_eq
tff(fact_3890_mod__mult__right__eq,axiom,
    ! [A: int,B: 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,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_right_eq
tff(fact_3891_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) ).

% mod_mult_right_eq
tff(fact_3892_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_left_eq
tff(fact_3893_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_left_eq
tff(fact_3894_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: 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,A),C)),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_left_eq
tff(fact_3895_mod__mult__left__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,C)),B),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) ).

% mod_mult_left_eq
tff(fact_3896_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),modulo364778990260209775nteger(A,B)) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ).

% mult_mod_right
tff(fact_3897_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ).

% mult_mod_right
tff(fact_3898_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] : aa_int_int(aa_int_fun_int_int(times_times_int,C),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,C),A)),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ).

% mult_mod_right
tff(fact_3899_mult__mod__right,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),modulo8411746178871703098atural(A,B)) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ).

% mult_mod_right
tff(fact_3900_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,B)),C) ).

% mod_mult_mult2
tff(fact_3901_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),C) ).

% mod_mult_mult2
tff(fact_3902_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),C) ).

% mod_mult_mult2
tff(fact_3903_mod__mult__mult2,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,B)),C) ).

% mod_mult_mult2
tff(fact_3904_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A3: code_integer,B: code_integer,B4: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(A3,C) )
     => ( ( modulo364778990260209775nteger(B,C) = modulo364778990260209775nteger(B4,C) )
       => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A3),B4),C) ) ) ) ).

% mod_mult_cong
tff(fact_3905_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A3: nat,B: nat,B4: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A3),C) )
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B4),C) )
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A3),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_3906_mod__mult__cong,axiom,
    ! [A: int,C: int,A3: int,B: int,B4: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A3),C) )
     => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B4),C) )
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A3),B4)),C) ) ) ) ).

% mod_mult_cong
tff(fact_3907_mod__mult__cong,axiom,
    ! [A: code_natural,C: code_natural,A3: code_natural,B: code_natural,B4: code_natural] :
      ( ( modulo8411746178871703098atural(A,C) = modulo8411746178871703098atural(A3,C) )
     => ( ( modulo8411746178871703098atural(B,C) = modulo8411746178871703098atural(B4,C) )
       => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A3),B4),C) ) ) ) ).

% mod_mult_cong
tff(fact_3908_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] : modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,C)),modulo364778990260209775nteger(B,C)),C) = modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ).

% mod_mult_eq
tff(fact_3909_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: 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,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ).

% mod_mult_eq
tff(fact_3910_mod__mult__eq,axiom,
    ! [A: int,C: int,B: 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,A),C)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ).

% mod_mult_eq
tff(fact_3911_mod__mult__eq,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] : modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,C)),modulo8411746178871703098atural(B,C)),C) = modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) ).

% mod_mult_eq
tff(fact_3912_dvd__triv__right,axiom,
    ! [A: assn,B: assn] : aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),A)) ).

% dvd_triv_right
tff(fact_3913_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) ).

% dvd_triv_right
tff(fact_3914_dvd__triv__right,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) ).

% dvd_triv_right
tff(fact_3915_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) ).

% dvd_triv_right
tff(fact_3916_dvd__triv__right,axiom,
    ! [A: code_natural,B: code_natural] : aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)) ).

% dvd_triv_right
tff(fact_3917_dvd__mult__right,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
     => aa_assn_o(dvd_dvd_assn(B),C) ) ).

% dvd_mult_right
tff(fact_3918_dvd__mult__right,axiom,
    ! [A: nat,B: 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,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ).

% dvd_mult_right
tff(fact_3919_dvd__mult__right,axiom,
    ! [A: int,B: 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,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ).

% dvd_mult_right
tff(fact_3920_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(dvd_dvd_Code_integer(B),C) ) ).

% dvd_mult_right
tff(fact_3921_dvd__mult__right,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C)
     => aa_Code_natural_o(dvd_dvd_Code_natural(B),C) ) ).

% dvd_mult_right
tff(fact_3922_mult__dvd__mono,axiom,
    ! [A: assn,B: assn,C: assn,D2: assn] :
      ( aa_assn_o(dvd_dvd_assn(A),B)
     => ( aa_assn_o(dvd_dvd_assn(C),D2)
       => aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),C)),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_3923_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_3924_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_3925_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(C),D2)
       => aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_3926_mult__dvd__mono,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural,D2: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),D2)
       => aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ).

% mult_dvd_mono
tff(fact_3927_dvd__triv__left,axiom,
    ! [A: assn,B: assn] : aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)) ).

% dvd_triv_left
tff(fact_3928_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) ).

% dvd_triv_left
tff(fact_3929_dvd__triv__left,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) ).

% dvd_triv_left
tff(fact_3930_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) ).

% dvd_triv_left
tff(fact_3931_dvd__triv__left,axiom,
    ! [A: code_natural,B: code_natural] : aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)) ).

% dvd_triv_left
tff(fact_3932_dvd__mult__left,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(dvd_dvd_assn(aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),B)),C)
     => aa_assn_o(dvd_dvd_assn(A),C) ) ).

% dvd_mult_left
tff(fact_3933_dvd__mult__left,axiom,
    ! [A: nat,B: 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,A),B)),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ).

% dvd_mult_left
tff(fact_3934_dvd__mult__left,axiom,
    ! [A: int,B: 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,A),B)),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ).

% dvd_mult_left
tff(fact_3935_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
     => aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ).

% dvd_mult_left
tff(fact_3936_dvd__mult__left,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ).

% dvd_mult_left
tff(fact_3937_dvd__mult2,axiom,
    ! [A: assn,B: assn,C: assn] :
      ( aa_assn_o(dvd_dvd_assn(A),B)
     => aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% dvd_mult2
tff(fact_3938_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult2
tff(fact_3939_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult2
tff(fact_3940_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
     => aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult2
tff(fact_3941_dvd__mult2,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ).

% dvd_mult2
tff(fact_3942_dvd__mult,axiom,
    ! [A: assn,C: assn,B: assn] :
      ( aa_assn_o(dvd_dvd_assn(A),C)
     => aa_assn_o(dvd_dvd_assn(A),aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),C)) ) ).

% dvd_mult
tff(fact_3943_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) ) ).

% dvd_mult
tff(fact_3944_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) ) ).

% dvd_mult
tff(fact_3945_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),C)
     => aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) ) ).

% dvd_mult
tff(fact_3946_dvd__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),C)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) ) ).

% dvd_mult
tff(fact_3947_dvd__def,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(dvd_dvd_assn(B),A)
    <=> ? [K4: assn] : A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K4) ) ).

% dvd_def
tff(fact_3948_dvd__def,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
    <=> ? [K4: nat] : A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K4) ) ).

% dvd_def
tff(fact_3949_dvd__def,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
    <=> ? [K4: int] : A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K4) ) ).

% dvd_def
tff(fact_3950_dvd__def,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
    <=> ? [K4: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K4) ) ).

% dvd_def
tff(fact_3951_dvd__def,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
    <=> ? [K4: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),K4) ) ).

% dvd_def
tff(fact_3952_dvdI,axiom,
    ! [A: assn,B: assn,K: assn] :
      ( ( A = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K) )
     => aa_assn_o(dvd_dvd_assn(B),A) ) ).

% dvdI
tff(fact_3953_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% dvdI
tff(fact_3954_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(times_times_int,B),K) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% dvdI
tff(fact_3955_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K) )
     => aa_Code_integer_o(dvd_dvd_Code_integer(B),A) ) ).

% dvdI
tff(fact_3956_dvdI,axiom,
    ! [A: code_natural,B: code_natural,K: code_natural] :
      ( ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),K) )
     => aa_Code_natural_o(dvd_dvd_Code_natural(B),A) ) ).

% dvdI
tff(fact_3957_dvdE,axiom,
    ! [B: assn,A: assn] :
      ( aa_assn_o(dvd_dvd_assn(B),A)
     => ~ ! [K2: assn] : A != aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,B),K2) ) ).

% dvdE
tff(fact_3958_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ~ ! [K2: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),K2) ) ).

% dvdE
tff(fact_3959_dvdE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ~ ! [K2: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,B),K2) ) ).

% dvdE
tff(fact_3960_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
     => ~ ! [K2: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),K2) ) ).

% dvdE
tff(fact_3961_dvdE,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
     => ~ ! [K2: code_natural] : A != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),K2) ) ).

% dvdE
tff(fact_3962_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
       => aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer) ) ) ).

% dvd_unit_imp_unit
tff(fact_3963_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat) ) ) ).

% dvd_unit_imp_unit
tff(fact_3964_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
       => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int) ) ) ).

% dvd_unit_imp_unit
tff(fact_3965_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => aa_Code_integer_o(dvd_dvd_Code_integer(B),A) ) ).

% unit_imp_dvd
tff(fact_3966_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A) ) ).

% unit_imp_dvd
tff(fact_3967_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A) ) ).

% unit_imp_dvd
tff(fact_3968_one__dvd,axiom,
    ! [A: code_integer] : aa_Code_integer_o(dvd_dvd_Code_integer(one_one_Code_integer),A) ).

% one_dvd
tff(fact_3969_one__dvd,axiom,
    ! [A: assn] : aa_assn_o(dvd_dvd_assn(one_one_assn),A) ).

% one_dvd
tff(fact_3970_one__dvd,axiom,
    ! [A: rat] : aa_rat_o(dvd_dvd_rat(one_one_rat),A) ).

% one_dvd
tff(fact_3971_one__dvd,axiom,
    ! [A: nat] : aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,one_one_nat),A) ).

% one_dvd
tff(fact_3972_one__dvd,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(dvd_dvd_int,one_one_int),A) ).

% one_dvd
tff(fact_3973_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
    <=> ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3974_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3975_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
    <=> ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3976_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
    <=> ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) = one_one_Code_natural ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_3977_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger(A,C) = modulo364778990260209775nteger(B,C) )
     => ~ ! [D3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D3)) ) ).

% mod_eqE
tff(fact_3978_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),C) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C) )
     => ~ ! [D3: int] : B != aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),D3)) ) ).

% mod_eqE
tff(fact_3979_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( 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,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),A))) ) ) ).

% pos_zmod_mult_2
tff(fact_3980_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),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,numeral_numeral_int(bit0(one))),B))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,B),one_one_int)),A))),one_one_int) ) ) ).

% neg_zmod_mult_2
tff(fact_3981_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_3982_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_3983_not__is__unit__0,axiom,
    ~ aa_Code_integer_o(dvd_dvd_Code_integer(zero_z3403309356797280102nteger),one_one_Code_integer) ).

% not_is_unit_0
tff(fact_3984_not__is__unit__0,axiom,
    ~ aa_Code_natural_o(dvd_dvd_Code_natural(zero_z2226904508553997617atural),one_one_Code_natural) ).

% not_is_unit_0
tff(fact_3985_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3986_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,B),A) = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3987_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3988_unit__mult__right__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) )
      <=> ( B = C ) ) ) ).

% unit_mult_right_cancel
tff(fact_3989_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3990_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = aa_int_int(aa_int_fun_int_int(times_times_int,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3991_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3992_unit__mult__left__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C) )
      <=> ( B = C ) ) ) ).

% unit_mult_left_cancel
tff(fact_3993_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),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,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3994_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),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,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3995_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3996_mult__unit__dvd__iff_H,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C)
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(B),C) ) ) ).

% mult_unit_dvd_iff'
tff(fact_3997_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3998_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_3999_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C))
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4000_dvd__mult__unit__iff_H,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% dvd_mult_unit_iff'
tff(fact_4001_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4002_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4003_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),C)
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4004_mult__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),C)
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% mult_unit_dvd_iff
tff(fact_4005_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4006_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4007_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B))
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4008_dvd__mult__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% dvd_mult_unit_iff
tff(fact_4009_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        & aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat) ) ) ).

% is_unit_mult_iff
tff(fact_4010_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        & aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int) ) ) ).

% is_unit_mult_iff
tff(fact_4011_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
        & aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer) ) ) ).

% is_unit_mult_iff
tff(fact_4012_is__unit__mult__iff,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)),one_one_Code_natural)
    <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
        & aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural) ) ) ).

% is_unit_mult_iff
tff(fact_4013_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(B,C)),A) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_4014_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4015_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),A)),C) ) ) ).

% dvd_div_mult
tff(fact_4016_dvd__div__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(B,C)),A) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A),C) ) ) ).

% dvd_div_mult
tff(fact_4017_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(C),B)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% div_mult_swap
tff(fact_4018_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4019_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% div_mult_swap
tff(fact_4020_div__mult__swap,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) ) ) ).

% div_mult_swap
tff(fact_4021_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(C),B)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
       => ( divide6298287555418463151nteger(A,divide6298287555418463151nteger(B,C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4022_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4023_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4024_div__div__eq__right,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,divide5121882707175180666atural(B,C)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),C) ) ) ) ).

% div_div_eq_right
tff(fact_4025_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)),A)
     => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4026_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4027_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),C)),A)
     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4028_dvd__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)),A)
     => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ).

% dvd_div_mult2_eq
tff(fact_4029_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B)
     => aa_Code_integer_o(dvd_dvd_Code_integer(A),divide6298287555418463151nteger(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_4030_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B)
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_4031_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B)
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) ) ).

% dvd_mult_imp_div
tff(fact_4032_dvd__mult__imp__div,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),B)
     => aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,C)) ) ).

% dvd_mult_imp_div
tff(fact_4033_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D2: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(D2),C)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),divide6298287555418463151nteger(C,D2)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4034_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D2: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,D2),C)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),D2)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4035_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D2: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),C)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),D2)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),aa_int_int(aa_int_fun_int_int(times_times_int,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4036_div__mult__div__if__dvd,axiom,
    ! [B: code_natural,A: code_natural,D2: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(D2),C)
       => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),divide5121882707175180666atural(C,D2)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),D2)) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_4037_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),divide6298287555418463151nteger(C,B))
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4038_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B))
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4039_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B))
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4040_dvd__div__unit__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(C,B))
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% dvd_div_unit_iff
tff(fact_4041_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(divide6298287555418463151nteger(A,B)),C)
      <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4042_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),B)),C)
      <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4043_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),B)),C)
      <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4044_div__unit__dvd__iff,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),C)
      <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),C) ) ) ).

% div_unit_dvd_iff
tff(fact_4045_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4046_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4047_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4048_unit__div__cancel,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(C,A) )
      <=> ( B = C ) ) ) ).

% unit_div_cancel
tff(fact_4049_mod2__eq__if,axiom,
    ! [A: code_integer] :
      modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) = $ite(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A),zero_z3403309356797280102nteger,one_one_Code_integer) ).

% mod2_eq_if
tff(fact_4050_mod2__eq__if,axiom,
    ! [A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A),zero_zero_nat,one_one_nat) ).

% mod2_eq_if
tff(fact_4051_mod2__eq__if,axiom,
    ! [A: int] :
      aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),zero_zero_int,one_one_int) ).

% mod2_eq_if
tff(fact_4052_mod2__eq__if,axiom,
    ! [A: code_natural] :
      modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) = $ite(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A),zero_z2226904508553997617atural,one_one_Code_natural) ).

% mod2_eq_if
tff(fact_4053_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
       => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != zero_z3403309356797280102nteger ) )
     => ~ ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
         => ( modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) != one_one_Code_integer ) ) ) ).

% parity_cases
tff(fact_4054_parity__cases,axiom,
    ! [A: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != zero_zero_nat ) )
     => ~ ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
         => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) != one_one_nat ) ) ) ).

% parity_cases
tff(fact_4055_parity__cases,axiom,
    ! [A: int] :
      ( ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) != zero_zero_int ) )
     => ~ ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
         => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) != one_one_int ) ) ) ).

% parity_cases
tff(fact_4056_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
       => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) != zero_z2226904508553997617atural ) )
     => ~ ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
         => ( modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) != one_one_Code_natural ) ) ) ).

% parity_cases
tff(fact_4057_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))),modulo364778990260209775nteger(A,B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4058_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: 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,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4059_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: 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,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4060_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(2)
tff(fact_4061_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B))),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4062_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: 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,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B))),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4063_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: 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,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B))),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4064_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B))),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),C) ).

% cancel_div_mod_rules(1)
tff(fact_4065_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] : A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B)) ).

% mod_div_decomp
tff(fact_4066_mod__div__decomp,axiom,
    ! [A: nat,B: nat] : A = 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,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ).

% mod_div_decomp
tff(fact_4067_mod__div__decomp,axiom,
    ! [A: int,B: int] : A = 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,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ).

% mod_div_decomp
tff(fact_4068_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] : A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B)) ).

% mod_div_decomp
tff(fact_4069_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)),modulo364778990260209775nteger(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4070_div__mult__mod__eq,axiom,
    ! [A: nat,B: 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,A),B)),B)),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4071_div__mult__mod__eq,axiom,
    ! [A: int,B: 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,A),B)),B)),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% div_mult_mod_eq
tff(fact_4072_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)),modulo8411746178871703098atural(A,B)) = A ).

% div_mult_mod_eq
tff(fact_4073_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4074_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4075_mod__div__mult__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4076_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)) = A ).

% mod_div_mult_eq
tff(fact_4077_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,B)),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4078_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4079_mod__mult__div__eq,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = A ).

% mod_mult_div_eq
tff(fact_4080_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,B)),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))) = A ).

% mod_mult_div_eq
tff(fact_4081_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))),modulo364778990260209775nteger(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4082_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4083_mult__div__mod__eq,axiom,
    ! [B: int,A: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = A ).

% mult_div_mod_eq
tff(fact_4084_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] : aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))),modulo8411746178871703098atural(A,B)) = A ).

% mult_div_mod_eq
tff(fact_4085_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] : divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C))),divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),modulo364778990260209775nteger(B,C)),C)) ).

% div_mult1_eq
tff(fact_4086_div__mult1__eq,axiom,
    ! [A: nat,B: 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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B),C))),C)) ).

% div_mult1_eq
tff(fact_4087_div__mult1__eq,axiom,
    ! [A: int,B: 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,A),B)),C) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B),C))),C)) ).

% div_mult1_eq
tff(fact_4088_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] : divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),divide5121882707175180666atural(B,C))),divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),modulo8411746178871703098atural(B,C)),C)) ).

% div_mult1_eq
tff(fact_4089_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B)) = modulo364778990260209775nteger(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4090_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4091_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_div_mult_eq_mod
tff(fact_4092_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B)) = modulo8411746178871703098atural(A,B) ).

% minus_div_mult_eq_mod
tff(fact_4093_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4094_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4095_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4096_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),B) ).

% minus_mod_eq_div_mult
tff(fact_4097_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,modulo364778990260209775nteger(A,B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4098_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4099_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)) ).

% minus_mod_eq_mult_div
tff(fact_4100_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,modulo8411746178871703098atural(A,B)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B)) ).

% minus_mod_eq_mult_div
tff(fact_4101_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] : minus_8373710615458151222nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),divide6298287555418463151nteger(A,B))) = modulo364778990260209775nteger(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4102_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B))) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4103_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ).

% minus_mult_div_eq_mod
tff(fact_4104_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] : minus_7197305767214868737atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),divide5121882707175180666atural(A,B))) = modulo8411746178871703098atural(A,B) ).

% minus_mult_div_eq_mod
tff(fact_4105_unity__coeff__ex,axiom,
    ! [Pa: fun_rat_o,L: rat] :
      ( ? [X4: rat] : aa_rat_o(Pa,aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,L),X4))
    <=> ? [X4: rat] :
          ( aa_rat_o(dvd_dvd_rat(L),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X4),zero_zero_rat))
          & aa_rat_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_4106_unity__coeff__ex,axiom,
    ! [Pa: fun_nat_o,L: nat] :
      ( ? [X4: nat] : aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,L),X4))
    <=> ? [X4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,L),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X4),zero_zero_nat))
          & aa_nat_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_4107_unity__coeff__ex,axiom,
    ! [Pa: fun_int_o,L: int] :
      ( ? [X4: int] : aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(times_times_int,L),X4))
    <=> ? [X4: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,L),aa_int_int(aa_int_fun_int_int(plus_plus_int,X4),zero_zero_int))
          & aa_int_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_4108_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_integer_o,L: code_integer] :
      ( ? [X4: code_integer] : aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,L),X4))
    <=> ? [X4: code_integer] :
          ( aa_Code_integer_o(dvd_dvd_Code_integer(L),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X4),zero_z3403309356797280102nteger))
          & aa_Code_integer_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_4109_unity__coeff__ex,axiom,
    ! [Pa: fun_Code_natural_o,L: code_natural] :
      ( ? [X4: code_natural] : aa_Code_natural_o(Pa,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,L),X4))
    <=> ? [X4: code_natural] :
          ( aa_Code_natural_o(dvd_dvd_Code_natural(L),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,X4),zero_z2226904508553997617atural))
          & aa_Code_natural_o(Pa,X4) ) ) ).

% unity_coeff_ex
tff(fact_4110_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] : B != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C3) ) ) ).

% unit_dvdE
tff(fact_4111_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] : B != aa_int_int(aa_int_fun_int_int(times_times_int,A),C3) ) ) ).

% unit_dvdE
tff(fact_4112_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] : B != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C3) ) ) ).

% unit_dvdE
tff(fact_4113_unit__dvdE,axiom,
    ! [A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [C3: code_natural] : B != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C3) ) ) ).

% unit_dvdE
tff(fact_4114_inf__period_I3_J,axiom,
    ! [D2: rat,D: rat,T3: rat] :
      ( aa_rat_o(dvd_dvd_rat(D2),D)
     => ! [X3: rat,K3: rat] :
          ( aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),T3))
        <=> aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D))),T3)) ) ) ).

% inf_period(3)
tff(fact_4115_inf__period_I3_J,axiom,
    ! [D2: int,D: int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int,K3: int] :
          ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))),T3)) ) ) ).

% inf_period(3)
tff(fact_4116_inf__period_I3_J,axiom,
    ! [D2: code_integer,D: code_integer,T3: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(D2),D)
     => ! [X3: code_integer,K3: code_integer] :
          ( aa_Code_integer_o(dvd_dvd_Code_integer(D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),T3))
        <=> aa_Code_integer_o(dvd_dvd_Code_integer(D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D))),T3)) ) ) ).

% inf_period(3)
tff(fact_4117_inf__period_I4_J,axiom,
    ! [D2: rat,D: rat,T3: rat] :
      ( aa_rat_o(dvd_dvd_rat(D2),D)
     => ! [X3: rat,K3: rat] :
          ( ~ aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X3),T3))
        <=> ~ aa_rat_o(dvd_dvd_rat(D2),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,X3),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K3),D))),T3)) ) ) ).

% inf_period(4)
tff(fact_4118_inf__period_I4_J,axiom,
    ! [D2: int,D: int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int,K3: int] :
          ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
        <=> ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),aa_int_int(aa_int_fun_int_int(times_times_int,K3),D))),T3)) ) ) ).

% inf_period(4)
tff(fact_4119_inf__period_I4_J,axiom,
    ! [D2: code_integer,D: code_integer,T3: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(D2),D)
     => ! [X3: code_integer,K3: code_integer] :
          ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X3),T3))
        <=> ~ aa_Code_integer_o(dvd_dvd_Code_integer(D2),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,minus_8373710615458151222nteger(X3,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K3),D))),T3)) ) ) ).

% inf_period(4)
tff(fact_4120_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
       => ( ( divide6298287555418463151nteger(B,A) = C )
        <=> ( B = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4121_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = C )
        <=> ( B = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4122_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
       => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = C )
        <=> ( B = aa_int_int(aa_int_fun_int_int(times_times_int,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4123_dvd__div__eq__mult,axiom,
    ! [A: code_natural,B: code_natural,C: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
       => ( ( divide5121882707175180666atural(B,A) = C )
        <=> ( B = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),A) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_4124_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
       => ( aa_Code_integer_o(dvd_dvd_Code_integer(divide6298287555418463151nteger(A,B)),C)
        <=> aa_Code_integer_o(dvd_dvd_Code_integer(A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4125_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C)
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4126_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C)
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4127_div__dvd__iff__mult,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( ( B != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(divide5121882707175180666atural(A,B)),C)
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B)) ) ) ) ).

% div_dvd_iff_mult
tff(fact_4128_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(C),B)
       => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),divide6298287555418463151nteger(B,C))
        <=> aa_Code_integer_o(dvd_dvd_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4129_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C))
        <=> aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4130_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),B)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C))
        <=> aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4131_dvd__div__iff__mult,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( ( C != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),B)
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),divide5121882707175180666atural(B,C))
        <=> aa_Code_natural_o(dvd_dvd_Code_natural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C)),B) ) ) ) ).

% dvd_div_iff_mult
tff(fact_4132_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( aa_Code_integer_o(dvd_dvd_Code_integer(A),B)
         => ( aa_Code_integer_o(dvd_dvd_Code_integer(C),D2)
           => ( ( divide6298287555418463151nteger(B,A) = divide6298287555418463151nteger(D2,C) )
            <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4133_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),B)
         => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,C),D2)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),A) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,D2),C) )
            <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4134_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),B)
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,C),D2)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,B),A) = aa_int_int(aa_int_fun_int_int(divide_divide_int,D2),C) )
            <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,B),C) = aa_int_int(aa_int_fun_int_int(times_times_int,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4135_dvd__div__div__eq__mult,axiom,
    ! [A: code_natural,C: code_natural,B: code_natural,D2: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( ( C != zero_z2226904508553997617atural )
       => ( aa_Code_natural_o(dvd_dvd_Code_natural(A),B)
         => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),D2)
           => ( ( divide5121882707175180666atural(B,A) = divide5121882707175180666atural(D2,C) )
            <=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),D2) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_4136_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = zero_z3403309356797280102nteger )
      <=> ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4137_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: 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,A),B) = zero_zero_nat )
      <=> ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4138_unit__div__eq__0__iff,axiom,
    ! [B: int,A: 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,A),B) = zero_zero_int )
      <=> ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4139_unit__div__eq__0__iff,axiom,
    ! [B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = zero_z2226904508553997617atural )
      <=> ( A = zero_z2226904508553997617atural ) ) ) ).

% unit_div_eq_0_iff
tff(fact_4140_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(C),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4141_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4142_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4143_is__unit__div__mult2__eq,axiom,
    ! [B: code_natural,C: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_4144_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(C),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),divide6298287555418463151nteger(B,C)) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_4145_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4146_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: 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,A),aa_int_int(aa_int_fun_int_int(divide_divide_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)),C) ) ) ).

% unit_div_mult_swap
tff(fact_4147_unit__div__mult__swap,axiom,
    ! [C: code_natural,A: code_natural,B: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),divide5121882707175180666atural(B,C)) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B),C) ) ) ).

% unit_div_mult_swap
tff(fact_4148_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,divide6298287555418463151nteger(A,B)),C) = divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),C),B) ) ) ).

% unit_div_commute
tff(fact_4149_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),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,A),B)),C) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4150_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),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,A),B)),C) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),C)),B) ) ) ).

% unit_div_commute
tff(fact_4151_unit__div__commute,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(A,B)),C) = divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),C),B) ) ) ).

% unit_div_commute
tff(fact_4152_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(C),one_one_Code_integer)
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),A)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = divide6298287555418463151nteger(divide6298287555418463151nteger(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_4153_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: 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,B),A)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4154_div__mult__unit2,axiom,
    ! [C: int,B: int,A: 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,B),A)
       => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C) ) ) ) ).

% div_mult_unit2
tff(fact_4155_div__mult__unit2,axiom,
    ! [C: code_natural,B: code_natural,A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(C),one_one_Code_natural)
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),A)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),C)) = divide5121882707175180666atural(divide5121882707175180666atural(A,B),C) ) ) ) ).

% div_mult_unit2
tff(fact_4156_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( ( A = divide6298287555418463151nteger(C,B) )
      <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4157_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B),one_one_nat)
     => ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),B) )
      <=> ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4158_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B),one_one_int)
     => ( ( A = aa_int_int(aa_int_fun_int_int(divide_divide_int,C),B) )
      <=> ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4159_unit__eq__div2,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( A = divide5121882707175180666atural(C,B) )
      <=> ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B) = C ) ) ) ).

% unit_eq_div2
tff(fact_4160_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
     => ( ( divide6298287555418463151nteger(A,B) = C )
      <=> ( A = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4161_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: 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,A),B) = C )
      <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4162_unit__eq__div1,axiom,
    ! [B: int,A: int,C: 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,A),B) = C )
      <=> ( A = aa_int_int(aa_int_fun_int_int(times_times_int,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4163_unit__eq__div1,axiom,
    ! [B: code_natural,A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
     => ( ( divide5121882707175180666atural(A,B) = C )
      <=> ( A = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B) ) ) ) ).

% unit_eq_div1
tff(fact_4164_is__unit__power__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(power_8256067586552552935nteger(A,N)),one_one_Code_integer)
    <=> ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4165_is__unit__power__iff,axiom,
    ! [A: int,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(A),N)),one_one_int)
    <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4166_is__unit__power__iff,axiom,
    ! [A: 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,A),N)),one_one_nat)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
        | ( N = zero_zero_nat ) ) ) ).

% is_unit_power_iff
tff(fact_4167_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: 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,I2),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,I2),N))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,one_one_nat),I2)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% power_dvd_imp_le
tff(fact_4168_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(A),one_one_Code_integer)
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( aa_Code_integer_o(dvd_dvd_Code_integer(B3),one_one_Code_integer)
               => ( ( divide6298287555418463151nteger(one_one_Code_integer,A) = B3 )
                 => ( ( divide6298287555418463151nteger(one_one_Code_integer,B3) = A )
                   => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B3) = one_one_Code_integer )
                     => ( divide6298287555418463151nteger(C,A) != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4169_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,A),one_one_nat)
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,B3),one_one_nat)
               => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),A) = B3 )
                 => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B3) = A )
                   => ( ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B3) = one_one_nat )
                     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,C),A) != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4170_is__unitE,axiom,
    ! [A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,A),one_one_int)
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,B3),one_one_int)
               => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),A) = B3 )
                 => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B3) = A )
                   => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,A),B3) = one_one_int )
                     => ( aa_int_int(aa_int_fun_int_int(divide_divide_int,C),A) != aa_int_int(aa_int_fun_int_int(times_times_int,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4171_is__unitE,axiom,
    ! [A: code_natural,C: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(A),one_one_Code_natural)
     => ~ ( ( A != zero_z2226904508553997617atural )
         => ! [B3: code_natural] :
              ( ( B3 != zero_z2226904508553997617atural )
             => ( aa_Code_natural_o(dvd_dvd_Code_natural(B3),one_one_Code_natural)
               => ( ( divide5121882707175180666atural(one_one_Code_natural,A) = B3 )
                 => ( ( divide5121882707175180666atural(one_one_Code_natural,B3) = A )
                   => ( ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B3) = one_one_Code_natural )
                     => ( divide5121882707175180666atural(C,A) != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,C),B3) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_4172_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4173_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),B)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4174_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4175_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),B)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_4176_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(B),one_one_Code_integer)
       => ( divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),A)) = divide6298287555418463151nteger(one_one_Code_integer,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4177_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),A)) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,one_one_nat),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4178_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),A)) = aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4179_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( A != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(B),one_one_Code_natural)
       => ( divide5121882707175180666atural(A,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,B),A)) = divide5121882707175180666atural(one_one_Code_natural,B) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_4180_evenE,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B3) ) ).

% evenE
tff(fact_4181_evenE,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B3) ) ).

% evenE
tff(fact_4182_evenE,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B3) ) ).

% evenE
tff(fact_4183_evenE,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ~ ! [B3: code_natural] : A != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),B3) ) ).

% evenE
tff(fact_4184_odd__one,axiom,
    ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),one_one_nat) ).

% odd_one
tff(fact_4185_odd__one,axiom,
    ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),one_one_int) ).

% odd_one
tff(fact_4186_odd__one,axiom,
    ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),one_one_Code_integer) ).

% odd_one
tff(fact_4187_odd__one,axiom,
    ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),one_one_Code_natural) ).

% odd_one
tff(fact_4188_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( aa_Code_integer_o(dvd_dvd_Code_integer(power_8256067586552552935nteger(X,M2)),power_8256067586552552935nteger(X,N))
      <=> ( aa_Code_integer_o(dvd_dvd_Code_integer(X),one_one_Code_integer)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).

% dvd_power_iff
tff(fact_4189_dvd__power__iff,axiom,
    ! [X: code_natural,M2: nat,N: nat] :
      ( ( X != zero_z2226904508553997617atural )
     => ( aa_Code_natural_o(dvd_dvd_Code_natural(power_7079662738309270450atural(X,M2)),power_7079662738309270450atural(X,N))
      <=> ( aa_Code_natural_o(dvd_dvd_Code_natural(X),one_one_Code_natural)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).

% dvd_power_iff
tff(fact_4190_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,aa_nat_int(power_power_int(X),M2)),aa_nat_int(power_power_int(X),N))
      <=> ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).

% dvd_power_iff
tff(fact_4191_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != 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,X),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N))
      <=> ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),one_one_nat)
          | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ) ).

% dvd_power_iff
tff(fact_4192_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_Code_integer ) )
     => aa_Code_integer_o(dvd_dvd_Code_integer(X),power_8256067586552552935nteger(X,N)) ) ).

% dvd_power
tff(fact_4193_dvd__power,axiom,
    ! [N: nat,X: rat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_rat ) )
     => aa_rat_o(dvd_dvd_rat(X),power_power_rat(X,N)) ) ).

% dvd_power
tff(fact_4194_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_int ) )
     => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),aa_nat_int(power_power_int(X),N)) ) ).

% dvd_power
tff(fact_4195_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)
        | ( X = one_one_nat ) )
     => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,X),N)) ) ).

% dvd_power
tff(fact_4196_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,N),M2)),M2)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
tff(fact_4197_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,M2),N)),M2)
      <=> ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
tff(fact_4198_set__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_se2793503036327961859nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% set_bit_Suc
tff(fact_4199_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se1617098188084679374atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one)))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se1617098188084679374atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))))) ).

% set_bit_Suc
tff(fact_4200_set__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,A),numeral_numeral_nat(bit0(one)))))) ).

% set_bit_Suc
tff(fact_4201_set__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se7879613467334960850it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))))) ).

% set_bit_Suc
tff(fact_4202_unset__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_se8260200283734997820nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_4203_unset__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se4205575877204974255it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,A),numeral_numeral_nat(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_4204_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se7083795435491715335atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one)))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se7083795435491715335atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_4205_unset__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se4203085406695923979it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))))) ).

% unset_bit_Suc
tff(fact_4206_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),C)
     => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),C)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,B),modulo364778990260209775nteger(divide6298287555418463151nteger(A,B),C))),modulo364778990260209775nteger(A,B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4207_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: 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,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),C)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B)),C))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4208_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: 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,A),aa_int_int(aa_int_fun_int_int(times_times_int,B),C)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),C))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B)) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_4209_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4210_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4211_even__two__times__div__two,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4212_even__two__times__div__two,axiom,
    ! [A: code_natural] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ( aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))) = A ) ) ).

% even_two_times_div_two
tff(fact_4213_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,zero_zero_nat)),M2)
     => ( 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,M2),N))),M2) = one_one_nat ) ) ).

% Suc_times_mod_eq
tff(fact_4214_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B))),B)
       => ( modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)) = modulo364778990260209775nteger(A,B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4215_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4216_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ).

% divmod_digit_0(2)
tff(fact_4217_oddE,axiom,
    ! [A: nat] :
      ( ~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)
     => ~ ! [B3: nat] : A != aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B3)),one_one_nat) ) ).

% oddE
tff(fact_4218_oddE,axiom,
    ! [A: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)
     => ~ ! [B3: int] : A != aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B3)),one_one_int) ) ).

% oddE
tff(fact_4219_oddE,axiom,
    ! [A: code_integer] :
      ( ~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A)
     => ~ ! [B3: code_integer] : A != aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B3)),one_one_Code_integer) ) ).

% oddE
tff(fact_4220_oddE,axiom,
    ! [A: code_natural] :
      ( ~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A)
     => ~ ! [B3: code_natural] : A != aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),B3)),one_one_Code_natural) ) ).

% oddE
tff(fact_4221_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,numeral_numeral_nat(bit0(one))),N),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% minus_one_power_iff
tff(fact_4222_minus__one__power__iff,axiom,
    ! [N: nat] :
      power_power_rat(uminus_uminus_rat(one_one_rat),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),one_one_rat,uminus_uminus_rat(one_one_rat)) ).

% minus_one_power_iff
tff(fact_4223_minus__one__power__iff,axiom,
    ! [N: nat] :
      power_8256067586552552935nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),N) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),N),one_one_Code_integer,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% minus_one_power_iff
tff(fact_4224_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B))),B)
       => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B))) = divide6298287555418463151nteger(A,B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4225_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B)
       => ( aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))) = aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4226_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),B)
       => ( aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))) = aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B) ) ) ) ).

% divmod_digit_0(1)
tff(fact_4227_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( modulo364778990260209775nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2)),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,modulo364778990260209775nteger(A,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4228_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(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,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4229_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2))),aa_nat_int(power_power_int(numeral_numeral_int(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,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4230_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_natural] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
     => ( modulo8411746178871703098atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2)),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(A,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),M2)))),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2)) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_4231_mod__double__modulus,axiom,
    ! [M2: code_integer,X: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),M2)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),X)
       => ( ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),M2)) = modulo364778990260209775nteger(X,M2) )
          | ( modulo364778990260209775nteger(X,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),M2)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(X,M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_4232_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),X)
       => ( ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2) )
          | ( aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M2)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_4233_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
       => ( ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),M2)) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2) )
          | ( aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),M2)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,X),M2)),M2) ) ) ) ) ).

% mod_double_modulus
tff(fact_4234_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),B)
       => ( aa_Code_integer_o(ord_le3102999989581377725nteger(B),modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)))
         => ( minus_8373710615458151222nteger(modulo364778990260209775nteger(A,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)),B) = modulo364778990260209775nteger(A,B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4235_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_zero_nat),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),B)
       => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)))
         => ( aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),B) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4236_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)))
         => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),B) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),B) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_4237_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).

% even_mask_div_iff'
tff(fact_4238_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).

% even_mask_div_iff'
tff(fact_4239_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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(numeral_numeral_int(bit0(one))),M2)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).

% even_mask_div_iff'
tff(fact_4240_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(minus_7197305767214868737atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2),one_one_Code_natural),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)))
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ).

% even_mask_div_iff'
tff(fact_4241_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2),one_one_Code_integer),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)))
    <=> ( ( power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N) = zero_z3403309356797280102nteger )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% even_mask_div_iff
tff(fact_4242_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(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,numeral_numeral_nat(bit0(one))),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)))
    <=> ( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N) = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% even_mask_div_iff
tff(fact_4243_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(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(numeral_numeral_int(bit0(one))),M2)),one_one_int)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)))
    <=> ( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) = zero_zero_int )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% even_mask_div_iff
tff(fact_4244_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(minus_7197305767214868737atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2),one_one_Code_natural),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)))
    <=> ( ( power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N) = zero_z2226904508553997617atural )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N) ) ) ).

% even_mask_div_iff
tff(fact_4245_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_4246_verit__le__mono__div,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_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,A2),N)),
                $ite(aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,B2),N) = zero_zero_nat,one_one_nat,zero_zero_nat))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,B2),N)) ) ) ).

% verit_le_mono_div
tff(fact_4247_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N: 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),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,A2),N)),
                $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,B2),N) = zero_zero_int,one_one_int,zero_zero_int))),aa_int_int(aa_int_fun_int_int(divide_divide_int,B2),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_4248_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_4249_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,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% one_div_minus_numeral
tff(fact_4250_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)),numeral_numeral_int(N)) = aa_int_int(uminus_uminus_int,adjust_div(unique5052692396658037445od_int(one,N))) ).

% minus_one_div_numeral
tff(fact_4251_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,numeral_numeral_int(N))) = aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(adjust_mod,numeral_numeral_int(N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N)))) ).

% one_mod_minus_numeral
tff(fact_4252_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)),numeral_numeral_int(N)) = aa_int_int(aa_int_fun_int_int(adjust_mod,numeral_numeral_int(N)),aa_Pro6799258434465214121nt_int(product_snd_int_int,unique5052692396658037445od_int(one,N))) ).

% minus_one_mod_numeral
tff(fact_4253_int__less__induct,axiom,
    ! [I2: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,I2),K)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_less_induct
tff(fact_4254_int__le__induct,axiom,
    ! [I2: int,K: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),K)
     => ( aa_int_o(Pa,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_le_induct
tff(fact_4255_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_4256_int__gr__induct,axiom,
    ! [K: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I2)
     => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,K),one_one_int))
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_int,K),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_gr_induct
tff(fact_4257_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_4258_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),M2)
     => ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
      <=> ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_4259_zsgn__def,axiom,
    ! [I2: int] :
      aa_int_int(sgn_sgn_int,I2) = $ite(
        I2 = zero_zero_int,
        zero_zero_int,
        $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),I2),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ) ).

% zsgn_def
tff(fact_4260_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_4261_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_4262_int__ge__induct,axiom,
    ! [K: int,I2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I2)
     => ( aa_int_o(Pa,K)
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_ge_induct
tff(fact_4263_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_4264_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_4265_int__induct,axiom,
    ! [Pa: fun_int_o,K: int,I2: int] :
      ( aa_int_o(Pa,K)
     => ( ! [I3: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),I3)
           => ( aa_int_o(Pa,I3)
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)) ) )
       => ( ! [I3: int] :
              ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),K)
             => ( aa_int_o(Pa,I3)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,I3),one_one_int)) ) )
         => aa_int_o(Pa,I2) ) ) ) ).

% int_induct
tff(fact_4266_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2 = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_4267_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(times_times_int,M2),N) = one_one_int )
    <=> ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2 = 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_4268_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_4269_flip__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_se1345352211410354436nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_4270_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se168947363167071951atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one)))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se168947363167071951atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_4271_flip__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(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,A),numeral_numeral_nat(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_4272_flip__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_se2159334234014336723it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))))) ).

% flip_bit_Suc
tff(fact_4273_signed__take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      bit_ri6519982836138164636nteger(N,A) = $ite(N = zero_zero_nat,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))))) ).

% signed_take_bit_rec
tff(fact_4274_signed__take__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(bit_ri631733984087533419it_int(N),A) = $ite(N = zero_zero_nat,aa_int_int(uminus_uminus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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,A),numeral_numeral_int(bit0(one))))))) ).

% signed_take_bit_rec
tff(fact_4275_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,A),one_one_int),B),product_Pair_int_int(Q2,R2))
       => 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,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),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,numeral_numeral_int(bit0(one))),R2)),one_one_int))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_4276_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
     => ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2))
       => 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,numeral_numeral_int(bit0(one))),A)),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),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,numeral_numeral_int(bit0(one))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_4277_signed__take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_ri6519982836138164636nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_4278_signed__take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(bit_ri631733984087533419it_int(N),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))))) ).

% signed_take_bit_Suc
tff(fact_4279_signed__take__bit__of__minus__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% signed_take_bit_of_minus_1
tff(fact_4280_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_4281_signed__take__bit__Suc__1,axiom,
    ! [N: nat] : bit_ri6519982836138164636nteger(aa_nat_nat(suc,N),one_one_Code_integer) = one_one_Code_integer ).

% signed_take_bit_Suc_1
tff(fact_4282_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_4283_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : bit_ri6519982836138164636nteger(numeral_numeral_nat(K),one_one_Code_integer) = one_one_Code_integer ).

% signed_take_bit_numeral_of_1
tff(fact_4284_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(K)),one_one_int) = one_one_int ).

% signed_take_bit_numeral_of_1
tff(fact_4285_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R2: int] :
      ( aa_Pro4592068669816069595_int_o(eucl_rel_int(A,B),product_Pair_int_int(Q2,R2))
     => ( ( B != zero_zero_int )
       => aa_Pro4592068669816069595_int_o(eucl_rel_int(aa_int_int(uminus_uminus_int,A),B),
            product_Pair_int_int($ite(R2 = 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(R2 = zero_zero_int,zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),R2)))) ) ) ).

% zminus1_lemma
tff(fact_4286_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,numeral_numeral_int(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,numeral_numeral_int(K))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_4287_divmod__BitM__2__eq,axiom,
    ! [M2: num] : unique5052692396658037445od_int(bitM(M2),bit0(one)) = product_Pair_int_int(aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(M2)),one_one_int),one_one_int) ).

% divmod_BitM_2_eq
tff(fact_4288_flip__bit__0,axiom,
    ! [A: code_integer] : bit_se1345352211410354436nteger(zero_zero_nat,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% flip_bit_0
tff(fact_4289_flip__bit__0,axiom,
    ! [A: code_natural] : bit_se168947363167071951atural(zero_zero_nat,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))) ).

% flip_bit_0
tff(fact_4290_flip__bit__0,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,zero_zero_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))) ).

% flip_bit_0
tff(fact_4291_flip__bit__0,axiom,
    ! [A: int] : aa_int_int(bit_se2159334234014336723it_int(zero_zero_nat),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))))) ).

% flip_bit_0
tff(fact_4292_take__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      bit_se1745604003318907178nteger(N,A) = $ite(N = zero_zero_nat,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))),numera6620942414471956472nteger(bit0(one)))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% take_bit_rec
tff(fact_4293_take__bit__rec,axiom,
    ! [N: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(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(bit_se2925701944663578781it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))))) ).

% take_bit_rec
tff(fact_4294_take__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      bit_se569199155075624693atural(N,A) = $ite(N = zero_zero_nat,zero_z2226904508553997617atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))),numera5444537566228673987atural(bit0(one)))),modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))))) ).

% take_bit_rec
tff(fact_4295_take__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(bit_se2923211474154528505it_int(N),A) = $ite(N = zero_zero_nat,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,aa_int_int(bit_se2923211474154528505it_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,A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))))) ).

% take_bit_rec
tff(fact_4296_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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),numeral_numeral_int(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(bit0(one))))))) ) ) ).

% and_int_unfold
tff(fact_4297_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo364778990260209775nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4298_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4299_one__mod__2__pow__eq,axiom,
    ! [N: nat] : modulo8411746178871703098atural(one_one_Code_natural,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4300_one__mod__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% one_mod_2_pow_eq
tff(fact_4301_upto__aux__rec,axiom,
    ! [I2: int,J: int,Js: list_int] :
      upto_aux(I2,J,Js) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),Js,upto_aux(I2,aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int),cons_int(J,Js))) ).

% upto_aux_rec
tff(fact_4302_push__bit__numeral__minus__1,axiom,
    ! [N: num] : bit_se7788150548672797655nteger(numeral_numeral_nat(N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),numeral_numeral_nat(N))) ).

% push_bit_numeral_minus_1
tff(fact_4303_push__bit__numeral__minus__1,axiom,
    ! [N: num] : aa_int_int(bit_se545348938243370406it_int(numeral_numeral_nat(N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),numeral_numeral_nat(N))) ).

% push_bit_numeral_minus_1
tff(fact_4304_of__bool__eq_I2_J,axiom,
    zero_n356916108424825756nteger($true) = one_one_Code_integer ).

% of_bool_eq(2)
tff(fact_4305_of__bool__eq_I2_J,axiom,
    zero_n2052037380579107095ol_rat($true) = one_one_rat ).

% of_bool_eq(2)
tff(fact_4306_of__bool__eq_I2_J,axiom,
    zero_n2687167440665602831ol_nat($true) = one_one_nat ).

% of_bool_eq(2)
tff(fact_4307_of__bool__eq_I2_J,axiom,
    zero_n2684676970156552555ol_int($true) = one_one_int ).

% of_bool_eq(2)
tff(fact_4308_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n356916108424825756nteger((Pa)) = one_one_Code_integer )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_4309_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n2052037380579107095ol_rat((Pa)) = one_one_rat )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_4310_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n2687167440665602831ol_nat((Pa)) = one_one_nat )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_4311_of__bool__eq__1__iff,axiom,
    ! [Pa: $o] :
      ( ( zero_n2684676970156552555ol_int((Pa)) = one_one_int )
    <=> (Pa) ) ).

% of_bool_eq_1_iff
tff(fact_4312_take__bit__Suc__1,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_Suc_1
tff(fact_4313_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),one_one_nat) = one_one_nat ).

% take_bit_Suc_1
tff(fact_4314_take__bit__Suc__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),one_one_int) = one_one_int ).

% take_bit_Suc_1
tff(fact_4315_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_n356916108424825756nteger((Pa))),one_one_Code_integer)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_4316_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_rat_o(ord_less_rat(zero_n2052037380579107095ol_rat((Pa))),one_one_rat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_4317_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_n2687167440665602831ol_nat((Pa))),one_one_nat)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_4318_of__bool__less__one__iff,axiom,
    ! [Pa: $o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_n2684676970156552555ol_int((Pa))),one_one_int)
    <=> ~ (Pa) ) ).

% of_bool_less_one_iff
tff(fact_4319_and_Oleft__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),A) = A ).

% and.left_neutral
tff(fact_4320_and_Oleft__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,one_one_int)),A) = A ).

% and.left_neutral
tff(fact_4321_and_Oright__neutral,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = A ).

% and.right_neutral
tff(fact_4322_and_Oright__neutral,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(uminus_uminus_int,one_one_int)) = A ).

% and.right_neutral
tff(fact_4323_bit_Oconj__one__right,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = X ).

% bit.conj_one_right
tff(fact_4324_bit_Oconj__one__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = X ).

% bit.conj_one_right
tff(fact_4325_of__bool__not__iff,axiom,
    ! [Pa: $o] : zero_n2052037380579107095ol_rat(~ (Pa)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,one_one_rat),zero_n2052037380579107095ol_rat((Pa))) ).

% of_bool_not_iff
tff(fact_4326_of__bool__not__iff,axiom,
    ! [Pa: $o] : zero_n356916108424825756nteger(~ (Pa)) = minus_8373710615458151222nteger(one_one_Code_integer,zero_n356916108424825756nteger((Pa))) ).

% of_bool_not_iff
tff(fact_4327_of__bool__not__iff,axiom,
    ! [Pa: $o] : zero_n2684676970156552555ol_int(~ (Pa)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,one_one_int),zero_n2684676970156552555ol_int((Pa))) ).

% of_bool_not_iff
tff(fact_4328_take__bit__numeral__1,axiom,
    ! [L: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(L),one_one_Code_integer) = one_one_Code_integer ).

% take_bit_numeral_1
tff(fact_4329_take__bit__numeral__1,axiom,
    ! [L: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),one_one_nat) = one_one_nat ).

% take_bit_numeral_1
tff(fact_4330_take__bit__numeral__1,axiom,
    ! [L: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),one_one_int) = one_one_int ).

% take_bit_numeral_1
tff(fact_4331_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_zero_nat )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4332_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger(N,one_one_Code_integer) = zero_z3403309356797280102nteger )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4333_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( bit_se569199155075624693atural(N,one_one_Code_natural) = zero_z2226904508553997617atural )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4334_take__bit__of__1__eq__0__iff,axiom,
    ! [N: nat] :
      ( ( aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_zero_int )
    <=> ( N = zero_zero_nat ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4335_and__numerals_I8_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit1(X))),one_one_Code_integer) = one_one_Code_integer ).

% and_numerals(8)
tff(fact_4336_and__numerals_I8_J,axiom,
    ! [X: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit1(X)),one_one_Code_natural) = one_one_Code_natural ).

% and_numerals(8)
tff(fact_4337_and__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit1(X))),one_one_int) = one_one_int ).

% and_numerals(8)
tff(fact_4338_and__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit1(X))),one_one_nat) = one_one_nat ).

% and_numerals(8)
tff(fact_4339_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),numera6620942414471956472nteger(bit1(Y))) = one_one_Code_integer ).

% and_numerals(2)
tff(fact_4340_and__numerals_I2_J,axiom,
    ! [Y: num] : bit_se2773287842338716102atural(one_one_Code_natural,numera5444537566228673987atural(bit1(Y))) = one_one_Code_natural ).

% and_numerals(2)
tff(fact_4341_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),numeral_numeral_int(bit1(Y))) = one_one_int ).

% and_numerals(2)
tff(fact_4342_and__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(bit1(Y))) = one_one_nat ).

% and_numerals(2)
tff(fact_4343_sgn__mult__self__eq,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),sgn_sgn_rat(A)) = zero_n2052037380579107095ol_rat(A != zero_zero_rat) ).

% sgn_mult_self_eq
tff(fact_4344_sgn__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = zero_n356916108424825756nteger(A != zero_z3403309356797280102nteger) ).

% sgn_mult_self_eq
tff(fact_4345_sgn__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(sgn_sgn_int,A)) = zero_n2684676970156552555ol_int(A != zero_zero_int) ).

% sgn_mult_self_eq
tff(fact_4346_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(Y))) = zero_z3403309356797280102nteger ).

% and_numerals(1)
tff(fact_4347_and__numerals_I1_J,axiom,
    ! [Y: num] : bit_se2773287842338716102atural(one_one_Code_natural,numera5444537566228673987atural(bit0(Y))) = zero_z2226904508553997617atural ).

% and_numerals(1)
tff(fact_4348_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),numeral_numeral_int(bit0(Y))) = zero_zero_int ).

% and_numerals(1)
tff(fact_4349_and__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),numeral_numeral_nat(bit0(Y))) = zero_zero_nat ).

% and_numerals(1)
tff(fact_4350_and__numerals_I5_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit0(X))),one_one_Code_integer) = zero_z3403309356797280102nteger ).

% and_numerals(5)
tff(fact_4351_and__numerals_I5_J,axiom,
    ! [X: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit0(X)),one_one_Code_natural) = zero_z2226904508553997617atural ).

% and_numerals(5)
tff(fact_4352_and__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit0(X))),one_one_int) = zero_zero_int ).

% and_numerals(5)
tff(fact_4353_and__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(X))),one_one_nat) = zero_zero_nat ).

% and_numerals(5)
tff(fact_4354_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% and_numerals(3)
tff(fact_4355_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% and_numerals(3)
tff(fact_4356_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit0(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% and_numerals(3)
tff(fact_4357_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% and_numerals(3)
tff(fact_4358_take__bit__of__1,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(N,one_one_Code_integer) = zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4359_take__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),one_one_nat) = zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4360_take__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),one_one_int) = zero_n2684676970156552555ol_int(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),N)) ).

% take_bit_of_1
tff(fact_4361_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,numeral_numeral_int(bit1(N)))),one_one_int) = one_one_int ).

% and_minus_numerals(6)
tff(fact_4362_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,numeral_numeral_int(bit1(N)))) = one_one_int ).

% and_minus_numerals(2)
tff(fact_4363_dbl__inc__simps_I3_J,axiom,
    aa_int_int(neg_nu5851722552734809277nc_int,one_one_int) = numeral_numeral_int(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_4364_dbl__inc__simps_I3_J,axiom,
    neg_nu5219082963157363817nc_rat(one_one_rat) = numeral_numeral_rat(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_4365_dbl__inc__simps_I3_J,axiom,
    neg_nu5831290666863070958nteger(one_one_Code_integer) = numera6620942414471956472nteger(bit1(one)) ).

% dbl_inc_simps(3)
tff(fact_4366_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% and_numerals(6)
tff(fact_4367_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% and_numerals(6)
tff(fact_4368_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit0(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% and_numerals(6)
tff(fact_4369_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% and_numerals(6)
tff(fact_4370_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% and_numerals(4)
tff(fact_4371_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% and_numerals(4)
tff(fact_4372_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit1(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% and_numerals(4)
tff(fact_4373_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit1(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% and_numerals(4)
tff(fact_4374_push__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se7788150548672797655nteger(aa_nat_nat(suc,N),A) = bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(bit0(one)))) ).

% push_bit_Suc
tff(fact_4375_push__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se6611745700429515170atural(aa_nat_nat(suc,N),A) = bit_se6611745700429515170atural(N,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),numera5444537566228673987atural(bit0(one)))) ).

% push_bit_Suc
tff(fact_4376_push__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(aa_nat_nat(suc,N)),A) = aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(bit0(one)))) ).

% push_bit_Suc
tff(fact_4377_push__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,aa_nat_nat(suc,N)),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(bit0(one)))) ).

% push_bit_Suc
tff(fact_4378_push__bit__of__1,axiom,
    ! [N: nat] : bit_se7788150548672797655nteger(N,one_one_Code_integer) = power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N) ).

% push_bit_of_1
tff(fact_4379_push__bit__of__1,axiom,
    ! [N: nat] : bit_se6611745700429515170atural(N,one_one_Code_natural) = power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N) ).

% push_bit_of_1
tff(fact_4380_push__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),one_one_int) = aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N) ).

% push_bit_of_1
tff(fact_4381_push__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N) ).

% push_bit_of_1
tff(fact_4382_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,numeral_numeral_int(bit0(N)))),one_one_int) = zero_zero_int ).

% and_minus_numerals(5)
tff(fact_4383_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,numeral_numeral_int(bit0(N)))) = zero_zero_int ).

% and_minus_numerals(1)
tff(fact_4384_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,numeral_numeral_int(bit1(one))) ).

% dbl_dec_simps(4)
tff(fact_4385_dbl__dec__simps_I4_J,axiom,
    neg_nu3179335615603231917ec_rat(uminus_uminus_rat(one_one_rat)) = uminus_uminus_rat(numeral_numeral_rat(bit1(one))) ).

% dbl_dec_simps(4)
tff(fact_4386_dbl__dec__simps_I4_J,axiom,
    neg_nu7757733837767384882nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(bit1(one))) ).

% dbl_dec_simps(4)
tff(fact_4387_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% and_numerals(7)
tff(fact_4388_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se2773287842338716102atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2773287842338716102atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% and_numerals(7)
tff(fact_4389_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit1(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% and_numerals(7)
tff(fact_4390_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit1(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% and_numerals(7)
tff(fact_4391_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4392_one__div__2__pow__eq,axiom,
    ! [N: nat] : 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,numeral_numeral_nat(bit0(one))),N)) = zero_n2687167440665602831ol_nat(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4393_one__div__2__pow__eq,axiom,
    ! [N: nat] : divide5121882707175180666atural(one_one_Code_natural,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4394_one__div__2__pow__eq,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(N = zero_zero_nat) ).

% one_div_2_pow_eq
tff(fact_4395_bits__1__div__exp,axiom,
    ! [N: nat] : divide6298287555418463151nteger(one_one_Code_integer,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4396_bits__1__div__exp,axiom,
    ! [N: nat] : 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,numeral_numeral_nat(bit0(one))),N)) = zero_n2687167440665602831ol_nat(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4397_bits__1__div__exp,axiom,
    ! [N: nat] : divide5121882707175180666atural(one_one_Code_natural,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = zero_n8403883297036319079atural(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4398_bits__1__div__exp,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,one_one_int),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = zero_n2684676970156552555ol_int(N = zero_zero_nat) ).

% bits_1_div_exp
tff(fact_4399_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,M2),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) ).

% take_bit_of_exp
tff(fact_4400_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : bit_se1745604003318907178nteger(M2,power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) ).

% take_bit_of_exp
tff(fact_4401_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : bit_se569199155075624693atural(M2,power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) ).

% take_bit_of_exp
tff(fact_4402_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] : aa_int_int(bit_se2923211474154528505it_int(M2),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),M2))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ).

% take_bit_of_exp
tff(fact_4403_take__bit__of__2,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(bit0(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),N))),numeral_numeral_nat(bit0(one))) ).

% take_bit_of_2
tff(fact_4404_take__bit__of__2,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(bit0(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),N))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_of_2
tff(fact_4405_take__bit__of__2,axiom,
    ! [N: nat] : bit_se569199155075624693atural(N,numera5444537566228673987atural(bit0(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),N))),numera5444537566228673987atural(bit0(one))) ).

% take_bit_of_2
tff(fact_4406_take__bit__of__2,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),numeral_numeral_int(bit0(one))) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(bit0(one))),N))),numeral_numeral_int(bit0(one))) ).

% take_bit_of_2
tff(fact_4407_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,numeral_numeral_int(bit1(V2))),numeral_numeral_int(bit0(W))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,numeral_numeral_int(V2)),numeral_numeral_int(W)))),one_one_int) ).

% zmod_numeral_Bit1
tff(fact_4408_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_ri631733984087533419it_int(aa_nat_nat(suc,N)),numeral_numeral_int(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),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_Suc_bit1
tff(fact_4409_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      zero_n2687167440665602831ol_nat(( (Pa)
        & (Qa) )) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat((Pa))),zero_n2687167440665602831ol_nat((Qa))) ).

% of_bool_conj
tff(fact_4410_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      zero_n356916108424825756nteger(( (Pa)
        & (Qa) )) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger((Pa))),zero_n356916108424825756nteger((Qa))) ).

% of_bool_conj
tff(fact_4411_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      zero_n8403883297036319079atural(( (Pa)
        & (Qa) )) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_n8403883297036319079atural((Pa))),zero_n8403883297036319079atural((Qa))) ).

% of_bool_conj
tff(fact_4412_of__bool__conj,axiom,
    ! [Pa: $o,Qa: $o] :
      zero_n2684676970156552555ol_int(( (Pa)
        & (Qa) )) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int((Pa))),zero_n2684676970156552555ol_int((Qa))) ).

% of_bool_conj
tff(fact_4413_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),B) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
    <=> ( ( A = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
        & ( B = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_4414_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),B) = aa_int_int(uminus_uminus_int,one_one_int) )
    <=> ( ( A = aa_int_int(uminus_uminus_int,one_one_int) )
        & ( B = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_4415_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_Code_integer_o(ord_le3102999989581377725nteger(zero_n356916108424825756nteger((Pa))),one_one_Code_integer) ).

% of_bool_less_eq_one
tff(fact_4416_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_rat_o(ord_less_eq_rat(zero_n2052037380579107095ol_rat((Pa))),one_one_rat) ).

% of_bool_less_eq_one
tff(fact_4417_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,zero_n2687167440665602831ol_nat((Pa))),one_one_nat) ).

% of_bool_less_eq_one
tff(fact_4418_of__bool__less__eq__one,axiom,
    ! [Pa: $o] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_n2684676970156552555ol_int((Pa))),one_one_int) ).

% of_bool_less_eq_one
tff(fact_4419_of__bool__def,axiom,
    ! [P4: $o] :
      zero_n2687167440665602831ol_nat((P4)) = $ite((P4),one_one_nat,zero_zero_nat) ).

% of_bool_def
tff(fact_4420_of__bool__def,axiom,
    ! [P4: $o] :
      zero_n2052037380579107095ol_rat((P4)) = $ite((P4),one_one_rat,zero_zero_rat) ).

% of_bool_def
tff(fact_4421_of__bool__def,axiom,
    ! [P4: $o] :
      zero_n356916108424825756nteger((P4)) = $ite((P4),one_one_Code_integer,zero_z3403309356797280102nteger) ).

% of_bool_def
tff(fact_4422_of__bool__def,axiom,
    ! [P4: $o] :
      zero_n8403883297036319079atural((P4)) = $ite((P4),one_one_Code_natural,zero_z2226904508553997617atural) ).

% of_bool_def
tff(fact_4423_of__bool__def,axiom,
    ! [P4: $o] :
      zero_n2684676970156552555ol_int((P4)) = $ite((P4),one_one_int,zero_zero_int) ).

% of_bool_def
tff(fact_4424_split__of__bool,axiom,
    ! [Pa: fun_nat_o,P4: $o] :
      ( aa_nat_o(Pa,zero_n2687167440665602831ol_nat((P4)))
    <=> ( ( (P4)
         => aa_nat_o(Pa,one_one_nat) )
        & ( ~ (P4)
         => aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_of_bool
tff(fact_4425_split__of__bool,axiom,
    ! [Pa: fun_rat_o,P4: $o] :
      ( aa_rat_o(Pa,zero_n2052037380579107095ol_rat((P4)))
    <=> ( ( (P4)
         => aa_rat_o(Pa,one_one_rat) )
        & ( ~ (P4)
         => aa_rat_o(Pa,zero_zero_rat) ) ) ) ).

% split_of_bool
tff(fact_4426_split__of__bool,axiom,
    ! [Pa: fun_Code_integer_o,P4: $o] :
      ( aa_Code_integer_o(Pa,zero_n356916108424825756nteger((P4)))
    <=> ( ( (P4)
         => aa_Code_integer_o(Pa,one_one_Code_integer) )
        & ( ~ (P4)
         => aa_Code_integer_o(Pa,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool
tff(fact_4427_split__of__bool,axiom,
    ! [Pa: fun_Code_natural_o,P4: $o] :
      ( aa_Code_natural_o(Pa,zero_n8403883297036319079atural((P4)))
    <=> ( ( (P4)
         => aa_Code_natural_o(Pa,one_one_Code_natural) )
        & ( ~ (P4)
         => aa_Code_natural_o(Pa,zero_z2226904508553997617atural) ) ) ) ).

% split_of_bool
tff(fact_4428_split__of__bool,axiom,
    ! [Pa: fun_int_o,P4: $o] :
      ( aa_int_o(Pa,zero_n2684676970156552555ol_int((P4)))
    <=> ( ( (P4)
         => aa_int_o(Pa,one_one_int) )
        & ( ~ (P4)
         => aa_int_o(Pa,zero_zero_int) ) ) ) ).

% split_of_bool
tff(fact_4429_split__of__bool__asm,axiom,
    ! [Pa: fun_nat_o,P4: $o] :
      ( aa_nat_o(Pa,zero_n2687167440665602831ol_nat((P4)))
    <=> ~ ( ( (P4)
            & ~ aa_nat_o(Pa,one_one_nat) )
          | ( ~ (P4)
            & ~ aa_nat_o(Pa,zero_zero_nat) ) ) ) ).

% split_of_bool_asm
tff(fact_4430_split__of__bool__asm,axiom,
    ! [Pa: fun_rat_o,P4: $o] :
      ( aa_rat_o(Pa,zero_n2052037380579107095ol_rat((P4)))
    <=> ~ ( ( (P4)
            & ~ aa_rat_o(Pa,one_one_rat) )
          | ( ~ (P4)
            & ~ aa_rat_o(Pa,zero_zero_rat) ) ) ) ).

% split_of_bool_asm
tff(fact_4431_split__of__bool__asm,axiom,
    ! [Pa: fun_Code_integer_o,P4: $o] :
      ( aa_Code_integer_o(Pa,zero_n356916108424825756nteger((P4)))
    <=> ~ ( ( (P4)
            & ~ aa_Code_integer_o(Pa,one_one_Code_integer) )
          | ( ~ (P4)
            & ~ aa_Code_integer_o(Pa,zero_z3403309356797280102nteger) ) ) ) ).

% split_of_bool_asm
tff(fact_4432_split__of__bool__asm,axiom,
    ! [Pa: fun_Code_natural_o,P4: $o] :
      ( aa_Code_natural_o(Pa,zero_n8403883297036319079atural((P4)))
    <=> ~ ( ( (P4)
            & ~ aa_Code_natural_o(Pa,one_one_Code_natural) )
          | ( ~ (P4)
            & ~ aa_Code_natural_o(Pa,zero_z2226904508553997617atural) ) ) ) ).

% split_of_bool_asm
tff(fact_4433_split__of__bool__asm,axiom,
    ! [Pa: fun_int_o,P4: $o] :
      ( aa_int_o(Pa,zero_n2684676970156552555ol_int((P4)))
    <=> ~ ( ( (P4)
            & ~ aa_int_o(Pa,one_one_int) )
          | ( ~ (P4)
            & ~ aa_int_o(Pa,zero_zero_int) ) ) ) ).

% split_of_bool_asm
tff(fact_4434_and_Ocomm__monoid__axioms,axiom,
    aa_Code_integer_o(comm_m1654649897431166824nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.comm_monoid_axioms
tff(fact_4435_and_Ocomm__monoid__axioms,axiom,
    aa_int_o(comm_monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.comm_monoid_axioms
tff(fact_4436_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_4437_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_nat(bit1(N)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,numeral_numeral_nat(N)),numeral_numeral_nat(N))),one_one_nat) ).

% numeral_Bit1
tff(fact_4438_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_int(bit1(N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(N)),numeral_numeral_int(N))),one_one_int) ).

% numeral_Bit1
tff(fact_4439_numeral__Bit1,axiom,
    ! [N: num] : numeral_numeral_rat(bit1(N)) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,numeral_numeral_rat(N)),numeral_numeral_rat(N))),one_one_rat) ).

% numeral_Bit1
tff(fact_4440_numeral__Bit1,axiom,
    ! [N: num] : numera6620942414471956472nteger(bit1(N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,numera6620942414471956472nteger(N)),numera6620942414471956472nteger(N))),one_one_Code_integer) ).

% numeral_Bit1
tff(fact_4441_numeral__Bit1,axiom,
    ! [N: num] : numera5444537566228673987atural(bit1(N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,numera5444537566228673987atural(N)),numera5444537566228673987atural(N))),one_one_Code_natural) ).

% numeral_Bit1
tff(fact_4442_and_Omonoid__axioms,axiom,
    aa_Code_integer_o(monoid_Code_integer(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.monoid_axioms
tff(fact_4443_and_Omonoid__axioms,axiom,
    aa_int_o(monoid_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.monoid_axioms
tff(fact_4444_and_Osemilattice__neutr__axioms,axiom,
    aa_Code_integer_o(semila6106414119144251899nteger(bit_se3949692690581998587nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% and.semilattice_neutr_axioms
tff(fact_4445_and_Osemilattice__neutr__axioms,axiom,
    aa_int_o(semila9079005292280841162tr_int(bit_se725231765392027082nd_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% and.semilattice_neutr_axioms
tff(fact_4446_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(bit1(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(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one)))),one_one_nat) ).

% take_bit_Suc_bit1
tff(fact_4447_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),numera6620942414471956472nteger(bit1(K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one)))),one_one_Code_integer) ).

% take_bit_Suc_bit1
tff(fact_4448_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N),numera5444537566228673987atural(bit1(K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(N,numera5444537566228673987atural(K))),numera5444537566228673987atural(bit0(one)))),one_one_Code_natural) ).

% take_bit_Suc_bit1
tff(fact_4449_take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),numeral_numeral_int(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),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_bit1
tff(fact_4450_power3__eq__cube,axiom,
    ! [A: assn] : aa_nat_assn(power_power_assn(A),numeral_numeral_nat(bit1(one))) = aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,A),A)),A) ).

% power3_eq_cube
tff(fact_4451_power3__eq__cube,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,A),numeral_numeral_nat(bit1(one))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),A)),A) ).

% power3_eq_cube
tff(fact_4452_power3__eq__cube,axiom,
    ! [A: int] : aa_nat_int(power_power_int(A),numeral_numeral_nat(bit1(one))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),A)),A) ).

% power3_eq_cube
tff(fact_4453_power3__eq__cube,axiom,
    ! [A: code_integer] : power_8256067586552552935nteger(A,numeral_numeral_nat(bit1(one))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A)),A) ).

% power3_eq_cube
tff(fact_4454_power3__eq__cube,axiom,
    ! [A: code_natural] : power_7079662738309270450atural(A,numeral_numeral_nat(bit1(one))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),A)),A) ).

% power3_eq_cube
tff(fact_4455_push__bit__double,axiom,
    ! [N: nat,A: code_integer] : bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),numera6620942414471956472nteger(bit0(one)))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se7788150548672797655nteger(N,A)),numera6620942414471956472nteger(bit0(one))) ).

% push_bit_double
tff(fact_4456_push__bit__double,axiom,
    ! [N: nat,A: code_natural] : bit_se6611745700429515170atural(N,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),numera5444537566228673987atural(bit0(one)))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se6611745700429515170atural(N,A)),numera5444537566228673987atural(bit0(one))) ).

% push_bit_double
tff(fact_4457_push__bit__double,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(aa_int_fun_int_int(times_times_int,A),numeral_numeral_int(bit0(one)))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se545348938243370406it_int(N),A)),numeral_numeral_int(bit0(one))) ).

% push_bit_double
tff(fact_4458_push__bit__double,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),numeral_numeral_nat(bit0(one)))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A)),numeral_numeral_nat(bit0(one))) ).

% push_bit_double
tff(fact_4459_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,numeral_numeral_int(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,numeral_numeral_int(inc(K))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_Suc_minus_bit1
tff(fact_4460_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),numeral_numeral_nat(bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4461_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),numera6620942414471956472nteger(bit0(K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4462_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : bit_se569199155075624693atural(aa_nat_nat(suc,N),numera5444537566228673987atural(bit0(K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(N,numera5444537566228673987atural(K))),numera5444537566228673987atural(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4463_take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(N),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% take_bit_Suc_bit0
tff(fact_4464_and__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),one_one_Code_integer) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% and_one_eq
tff(fact_4465_and__one__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(A,one_one_Code_natural) = modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) ).

% and_one_eq
tff(fact_4466_and__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) ).

% and_one_eq
tff(fact_4467_and__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) ).

% and_one_eq
tff(fact_4468_one__and__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,one_one_Code_integer),A) = modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one))) ).

% one_and_eq
tff(fact_4469_one__and__eq,axiom,
    ! [A: code_natural] : bit_se2773287842338716102atural(one_one_Code_natural,A) = modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one))) ).

% one_and_eq
tff(fact_4470_one__and__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one))) ).

% one_and_eq
tff(fact_4471_one__and__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one))) ).

% one_and_eq
tff(fact_4472_numeral__BitM,axiom,
    ! [N: num] : numeral_numeral_int(bitM(N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(bit0(N))),one_one_int) ).

% numeral_BitM
tff(fact_4473_numeral__BitM,axiom,
    ! [N: num] : numeral_numeral_rat(bitM(N)) = aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,numeral_numeral_rat(bit0(N))),one_one_rat) ).

% numeral_BitM
tff(fact_4474_numeral__BitM,axiom,
    ! [N: num] : numera6620942414471956472nteger(bitM(N)) = minus_8373710615458151222nteger(numera6620942414471956472nteger(bit0(N)),one_one_Code_integer) ).

% numeral_BitM
tff(fact_4475_push__bit__eq__mult,axiom,
    ! [N: nat,A: code_integer] : bit_se7788150548672797655nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) ).

% push_bit_eq_mult
tff(fact_4476_push__bit__eq__mult,axiom,
    ! [N: nat,A: code_natural] : bit_se6611745700429515170atural(N,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,A),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) ).

% push_bit_eq_mult
tff(fact_4477_push__bit__eq__mult,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se545348938243370406it_int(N),A) = aa_int_int(aa_int_fun_int_int(times_times_int,A),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) ).

% push_bit_eq_mult
tff(fact_4478_push__bit__eq__mult,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) ).

% push_bit_eq_mult
tff(fact_4479_bits__induct,axiom,
    ! [Pa: fun_Code_integer_o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( divide6298287555418463151nteger(A4,numera6620942414471956472nteger(bit0(one))) = A4 )
         => aa_Code_integer_o(Pa,A4) )
     => ( ! [A4: code_integer,B3: $o] :
            ( aa_Code_integer_o(Pa,A4)
           => ( ( divide6298287555418463151nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger((B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),A4)),numera6620942414471956472nteger(bit0(one))) = A4 )
             => aa_Code_integer_o(Pa,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,zero_n356916108424825756nteger((B3))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),A4))) ) )
       => aa_Code_integer_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4480_bits__induct,axiom,
    ! [Pa: fun_nat_o,A: nat] :
      ( ! [A4: nat] :
          ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A4),numeral_numeral_nat(bit0(one))) = A4 )
         => aa_nat_o(Pa,A4) )
     => ( ! [A4: nat,B3: $o] :
            ( aa_nat_o(Pa,A4)
           => ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A4))),numeral_numeral_nat(bit0(one))) = A4 )
             => aa_nat_o(Pa,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,zero_n2687167440665602831ol_nat((B3))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),A4))) ) )
       => aa_nat_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4481_bits__induct,axiom,
    ! [Pa: fun_Code_natural_o,A: code_natural] :
      ( ! [A4: code_natural] :
          ( ( divide5121882707175180666atural(A4,numera5444537566228673987atural(bit0(one))) = A4 )
         => aa_Code_natural_o(Pa,A4) )
     => ( ! [A4: code_natural,B3: $o] :
            ( aa_Code_natural_o(Pa,A4)
           => ( ( divide5121882707175180666atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural((B3))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),A4)),numera5444537566228673987atural(bit0(one))) = A4 )
             => aa_Code_natural_o(Pa,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,zero_n8403883297036319079atural((B3))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),A4))) ) )
       => aa_Code_natural_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4482_bits__induct,axiom,
    ! [Pa: fun_int_o,A: int] :
      ( ! [A4: int] :
          ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A4),numeral_numeral_int(bit0(one))) = A4 )
         => aa_int_o(Pa,A4) )
     => ( ! [A4: int,B3: $o] :
            ( aa_int_o(Pa,A4)
           => ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int((B3))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A4))),numeral_numeral_int(bit0(one))) = A4 )
             => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,zero_n2684676970156552555ol_int((B3))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),A4))) ) )
       => aa_int_o(Pa,A) ) ) ).

% bits_induct
tff(fact_4483_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(numeral_numeral_int(bit0(one))),N)) ).

% push_bit_minus_one
tff(fact_4484_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : modulo364778990260209775nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,zero_n356916108424825756nteger(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2)) ).

% exp_mod_exp
tff(fact_4485_exp__mod__exp,axiom,
    ! [M2: 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,numeral_numeral_nat(bit0(one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2)) ).

% exp_mod_exp
tff(fact_4486_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : modulo8411746178871703098atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,zero_n8403883297036319079atural(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2)) ).

% exp_mod_exp
tff(fact_4487_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] : aa_int_int(aa_int_fun_int_int(modulo_modulo_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,zero_n2684676970156552555ol_int(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M2),N))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2)) ).

% exp_mod_exp
tff(fact_4488_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(numeral_numeral_int(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_4489_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),aa_nat_nat(suc,N)),one_one_Code_integer) ).

% take_bit_Suc_minus_1_eq
tff(fact_4490_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(suc,N))),one_one_int) ).

% take_bit_Suc_minus_1_eq
tff(fact_4491_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] : bit_se1745604003318907178nteger(aa_nat_nat(suc,N),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(N,divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))),numera6620942414471956472nteger(bit0(one)))),modulo364778990260209775nteger(A,numera6620942414471956472nteger(bit0(one)))) ).

% take_bit_Suc
tff(fact_4492_take__bit__Suc,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,aa_nat_nat(suc,N)),A) = 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(bit_se2925701944663578781it_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,A),numeral_numeral_nat(bit0(one)))) ).

% take_bit_Suc
tff(fact_4493_take__bit__Suc,axiom,
    ! [N: nat,A: code_natural] : bit_se569199155075624693atural(aa_nat_nat(suc,N),A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(N,divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))),numera5444537566228673987atural(bit0(one)))),modulo8411746178871703098atural(A,numera5444537566228673987atural(bit0(one)))) ).

% take_bit_Suc
tff(fact_4494_take__bit__Suc,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2923211474154528505it_int(aa_nat_nat(suc,N)),A) = 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(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))))),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,A),numeral_numeral_int(bit0(one)))) ).

% take_bit_Suc
tff(fact_4495_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(K),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),numeral_numeral_nat(K)),one_one_Code_integer) ).

% take_bit_numeral_minus_1_eq
tff(fact_4496_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(K)),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),numeral_numeral_nat(K))),one_one_int) ).

% take_bit_numeral_minus_1_eq
tff(fact_4497_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))) = A )
     => ( bit_se1745604003318907178nteger(N,A) = $ite(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A),zero_z3403309356797280102nteger,minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N),one_one_Code_integer)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4498_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))) = A )
     => ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,N),A) = $ite(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A),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,numeral_numeral_nat(bit0(one))),N)),one_one_nat)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4499_stable__imp__take__bit__eq,axiom,
    ! [A: code_natural,N: nat] :
      ( ( divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))) = A )
     => ( bit_se569199155075624693atural(N,A) = $ite(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A),zero_z2226904508553997617atural,minus_7197305767214868737atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N),one_one_Code_natural)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4500_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one))) = A )
     => ( aa_int_int(bit_se2923211474154528505it_int(N),A) = $ite(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),zero_zero_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)),one_one_int)) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4501_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      divide6298287555418463151nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,
          zero_n356916108424825756nteger(( ( power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),M2) != zero_z3403309356797280102nteger )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4502_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2)),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),N)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,
          zero_n2687167440665602831ol_nat(( ( aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),M2) != zero_zero_nat )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4503_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      divide5121882707175180666atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,
          zero_n8403883297036319079atural(( ( power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),M2) != zero_z2226904508553997617atural )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4504_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2)),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),N)) = aa_int_int(aa_int_fun_int_int(times_times_int,
          zero_n2684676970156552555ol_int(( ( aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),M2) != zero_zero_int )
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2) ))),aa_nat_int(power_power_int(numeral_numeral_int(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),N))) ).

% exp_div_exp_eq
tff(fact_4505_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(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(pred_numeral(L)),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))),one_one_int))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_4506_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_ri631733984087533419it_int(numeral_numeral_nat(L)),numeral_numeral_int(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(pred_numeral(L)),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% signed_take_bit_numeral_bit1
tff(fact_4507_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(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(pred_numeral(L)),aa_int_int(uminus_uminus_int,numeral_numeral_int(inc(K))))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_numeral_minus_bit1
tff(fact_4508_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
        ( member_int2(K,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
        & member_int2(L,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
        aa_int_int(uminus_uminus_int,
          zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)
            & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),
        aa_int_int(aa_int_fun_int_int(plus_plus_int,
            zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)
              & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(bit0(one)))))) ) ).

% and_int.simps
tff(fact_4509_and__int_Oelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( Y = $ite(
            ( member_int2(X,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int2(Xa,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.elims
tff(fact_4510_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(bit1(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(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one)))),one_one_nat) ).

% take_bit_numeral_bit1
tff(fact_4511_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(L),numera6620942414471956472nteger(bit1(K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(pred_numeral(L),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one)))),one_one_Code_integer) ).

% take_bit_numeral_bit1
tff(fact_4512_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : bit_se569199155075624693atural(numeral_numeral_nat(L),numera5444537566228673987atural(bit1(K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(pred_numeral(L),numera5444537566228673987atural(K))),numera5444537566228673987atural(bit0(one)))),one_one_Code_natural) ).

% take_bit_numeral_bit1
tff(fact_4513_take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),numeral_numeral_int(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(pred_numeral(L)),numeral_numeral_int(K))),numeral_numeral_int(bit0(one)))),one_one_int) ).

% take_bit_numeral_bit1
tff(fact_4514_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))),nil_int) ).

% upto_rec_numeral(4)
tff(fact_4515_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),numeral_numeral_int(N)),cons_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(M2)),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),one_one_int)),numeral_numeral_int(N))),nil_int) ).

% upto_rec_numeral(3)
tff(fact_4516_Int__insert__right__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(A,A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4517_Int__insert__right__if1,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( member_int2(A,A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int2(A,B2)) = insert_int2(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4518_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4519_Int__insert__right__if1,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( member_nat2(A,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ) ) ).

% Int_insert_right_if1
tff(fact_4520_Int__insert__right__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ member7544390268018397336_nat_o(A,A2)
     => ( inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = inf_in3376754634429691197_nat_o(A2,B2) ) ) ).

% Int_insert_right_if0
tff(fact_4521_Int__insert__right__if0,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ~ member_int2(A,A2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int2(A,B2)) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_4522_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,A2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_4523_Int__insert__right__if0,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ~ member_nat2(A,A2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ).

% Int_insert_right_if0
tff(fact_4524_insert__inter__insert,axiom,
    ! [A: int,A2: set_int,B2: set_int] : aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,A2)),insert_int2(A,B2)) = insert_int2(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ).

% insert_inter_insert
tff(fact_4525_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),insert8211810215607154385at_nat(A,B2)) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ).

% insert_inter_insert
tff(fact_4526_insert__inter__insert,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] : aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),insert_nat2(A,B2)) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% insert_inter_insert
tff(fact_4527_Int__insert__left__if1,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( member7544390268018397336_nat_o(A,C2)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4528_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( member_int2(A,C2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,B2)),C2) = insert_int2(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4529_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( member8440522571783428010at_nat(A,C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4530_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( member_nat2(A,C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C2) = insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ) ) ).

% Int_insert_left_if1
tff(fact_4531_Int__insert__left__if0,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,C2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ~ member7544390268018397336_nat_o(A,C2)
     => ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = inf_in3376754634429691197_nat_o(B2,C2) ) ) ).

% Int_insert_left_if0
tff(fact_4532_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B2: set_int] :
      ( ~ member_int2(A,C2)
     => ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,B2)),C2) = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4533_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ member8440522571783428010at_nat(A,C2)
     => ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4534_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B2: set_nat] :
      ( ~ member_nat2(A,C2)
     => ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C2) = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2) ) ) ).

% Int_insert_left_if0
tff(fact_4535_insert__disjoint_I1_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ member7544390268018397336_nat_o(A,B2)
        & ( inf_in3376754634429691197_nat_o(A2,B2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% insert_disjoint(1)
tff(fact_4536_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4537_insert__disjoint_I1_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),B2) = bot_bot_set_nat )
    <=> ( ~ member_nat2(A,B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
tff(fact_4538_insert__disjoint_I1_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,A2)),B2) = bot_bot_set_int )
    <=> ( ~ member_int2(A,B2)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
tff(fact_4539_insert__disjoint_I2_J,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,A2),B2) )
    <=> ( ~ member7544390268018397336_nat_o(A,B2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4540_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,A2)),B2) )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4541_insert__disjoint_I2_J,axiom,
    ! [A: nat,A2: set_nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,A2)),B2) )
    <=> ( ~ member_nat2(A,B2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4542_insert__disjoint_I2_J,axiom,
    ! [A: int,A2: set_int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,A2)),B2) )
    <=> ( ~ member_int2(A,B2)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ) ).

% insert_disjoint(2)
tff(fact_4543_disjoint__insert_I1_J,axiom,
    ! [B2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,A2: set_fu3555266933746557007_nat_o] :
      ( ( inf_in3376754634429691197_nat_o(B2,insert4923026906511862463_nat_o(A,A2)) = bot_bo8632255151191677115_nat_o )
    <=> ( ~ member7544390268018397336_nat_o(A,B2)
        & ( inf_in3376754634429691197_nat_o(B2,A2) = bot_bo8632255151191677115_nat_o ) ) ) ).

% disjoint_insert(1)
tff(fact_4544_disjoint__insert_I1_J,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),insert8211810215607154385at_nat(A,A2)) = bot_bo2099793752762293965at_nat )
    <=> ( ~ member8440522571783428010at_nat(A,B2)
        & ( aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),A2) = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4545_disjoint__insert_I1_J,axiom,
    ! [B2: set_nat,A: nat,A2: set_nat] :
      ( ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),insert_nat2(A,A2)) = bot_bot_set_nat )
    <=> ( ~ member_nat2(A,B2)
        & ( aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),A2) = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
tff(fact_4546_disjoint__insert_I1_J,axiom,
    ! [B2: set_int,A: int,A2: set_int] :
      ( ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),insert_int2(A,A2)) = bot_bot_set_int )
    <=> ( ~ member_int2(A,B2)
        & ( aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),A2) = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
tff(fact_4547_disjoint__insert_I2_J,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,B: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      ( ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(B,B2)) )
    <=> ( ~ member7544390268018397336_nat_o(B,A2)
        & ( bot_bo8632255151191677115_nat_o = inf_in3376754634429691197_nat_o(A2,B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4548_disjoint__insert_I2_J,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(B,B2)) )
    <=> ( ~ member8440522571783428010at_nat(B,A2)
        & ( bot_bo2099793752762293965at_nat = aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4549_disjoint__insert_I2_J,axiom,
    ! [A2: set_nat,B: nat,B2: set_nat] :
      ( ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(B,B2)) )
    <=> ( ~ member_nat2(B,A2)
        & ( bot_bot_set_nat = aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4550_disjoint__insert_I2_J,axiom,
    ! [A2: set_int,B: int,B2: set_int] :
      ( ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int2(B,B2)) )
    <=> ( ~ member_int2(B,A2)
        & ( bot_bot_set_int = aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2) ) ) ) ).

% disjoint_insert(2)
tff(fact_4551_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( aa_int_list_int(upto(I2),J) = nil_int )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil
tff(fact_4552_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int = aa_int_list_int(upto(I2),J) )
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2) ) ).

% upto_Nil2
tff(fact_4553_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2)
     => ( aa_int_list_int(upto(I2),J) = nil_int ) ) ).

% upto_empty
tff(fact_4554_upto__single,axiom,
    ! [I2: int] : aa_int_list_int(upto(I2),I2) = cons_int(I2,nil_int) ).

% upto_single
tff(fact_4555_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)),numeral_numeral_nat(bit1(Y))) = one_one_nat ).

% and_nat_numerals(2)
tff(fact_4556_and__nat__numerals_I4_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,numeral_numeral_nat(bit1(X))),aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% and_nat_numerals(4)
tff(fact_4557_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(numeral_numeral_int(M2)),numeral_numeral_int(N)) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),numeral_numeral_int(N)),cons_int(numeral_numeral_int(M2),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int)),numeral_numeral_int(N))),nil_int) ).

% upto_rec_numeral(1)
tff(fact_4558_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      aa_int_list_int(upto(numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(M2)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N))),cons_int(numeral_numeral_int(M2),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,numeral_numeral_int(M2)),one_one_int)),aa_int_int(uminus_uminus_int,numeral_numeral_int(N)))),nil_int) ).

% upto_rec_numeral(2)
tff(fact_4559_Int__insert__right,axiom,
    ! [A2: set_fu3555266933746557007_nat_o,A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(A2,insert4923026906511862463_nat_o(A,B2)) = $ite(member7544390268018397336_nat_o(A,A2),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(A2,B2)),inf_in3376754634429691197_nat_o(A2,B2)) ).

% Int_insert_right
tff(fact_4560_Int__insert__right,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),insert_int2(A,B2)) = $ite(member_int2(A,A2),insert_int2(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,A2),B2)) ).

% Int_insert_right
tff(fact_4561_Int__insert__right,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),insert8211810215607154385at_nat(A,B2)) = $ite(member8440522571783428010at_nat(A,A2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,A2),B2)) ).

% Int_insert_right
tff(fact_4562_Int__insert__right,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),insert_nat2(A,B2)) = $ite(member_nat2(A,A2),insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,A2),B2)) ).

% Int_insert_right
tff(fact_4563_Int__insert__left,axiom,
    ! [A: fun_Pr6460944699146269935_nat_o,B2: set_fu3555266933746557007_nat_o,C2: set_fu3555266933746557007_nat_o] :
      inf_in3376754634429691197_nat_o(insert4923026906511862463_nat_o(A,B2),C2) = $ite(member7544390268018397336_nat_o(A,C2),insert4923026906511862463_nat_o(A,inf_in3376754634429691197_nat_o(B2,C2)),inf_in3376754634429691197_nat_o(B2,C2)) ).

% Int_insert_left
tff(fact_4564_Int__insert__left,axiom,
    ! [A: int,B2: set_int,C2: set_int] :
      aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,insert_int2(A,B2)),C2) = $ite(member_int2(A,C2),insert_int2(A,aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)),aa_set_int_set_int(aa_set4234024024286713313et_int(inf_inf_set_int,B2),C2)) ).

% Int_insert_left
tff(fact_4565_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,insert8211810215607154385at_nat(A,B2)),C2) = $ite(member8440522571783428010at_nat(A,C2),insert8211810215607154385at_nat(A,aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)),aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,B2),C2)) ).

% Int_insert_left
tff(fact_4566_Int__insert__left,axiom,
    ! [A: nat,B2: set_nat,C2: set_nat] :
      aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,insert_nat2(A,B2)),C2) = $ite(member_nat2(A,C2),insert_nat2(A,aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)),aa_set_nat_set_nat(aa_set2309701074089104077et_nat(inf_inf_set_nat,B2),C2)) ).

% Int_insert_left
tff(fact_4567_upto__code,axiom,
    ! [I2: int,J: int] : aa_int_list_int(upto(I2),J) = upto_aux(I2,J,nil_int) ).

% upto_code
tff(fact_4568_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(K)),one_one_nat) ).

% pred_numeral_def
tff(fact_4569_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( aa_int_list_int(upto(X),Xa) = Y )
     => ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),Xa)),nil_int) ) ) ).

% upto.elims
tff(fact_4570_upto_Osimps,axiom,
    ! [I2: int,J: int] :
      aa_int_list_int(upto(I2),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J)),nil_int) ).

% upto.simps
tff(fact_4571_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_list_int(upto(I2),J) = cons_int(I2,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J)) ) ) ).

% upto_rec1
tff(fact_4572_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,numeral_numeral_nat(L)),numeral_numeral_nat(bit0(K))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se2925701944663578781it_nat,pred_numeral(L)),numeral_numeral_nat(K))),numeral_numeral_nat(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_4573_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_se1745604003318907178nteger(numeral_numeral_nat(L),numera6620942414471956472nteger(bit0(K))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,bit_se1745604003318907178nteger(pred_numeral(L),numera6620942414471956472nteger(K))),numera6620942414471956472nteger(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_4574_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : bit_se569199155075624693atural(numeral_numeral_nat(L),numera5444537566228673987atural(bit0(K))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,bit_se569199155075624693atural(pred_numeral(L),numera5444537566228673987atural(K))),numera5444537566228673987atural(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_4575_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa_int_int(bit_se2923211474154528505it_int(numeral_numeral_nat(L)),numeral_numeral_int(bit0(K))) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(bit_se2923211474154528505it_int(pred_numeral(L)),numeral_numeral_int(K))),numeral_numeral_int(bit0(one))) ).

% take_bit_numeral_bit0
tff(fact_4576_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K,L))
     => ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,K),L) = $ite(
            ( member_int2(K,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
            & member_int2(L,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
            aa_int_int(uminus_uminus_int,
              zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)
                & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),
            aa_int_int(aa_int_fun_int_int(plus_plus_int,
                zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),K)
                  & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),L) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(bit0(one)))))) ) ) ) ).

% and_int.psimps
tff(fact_4577_and__int_Opelims,axiom,
    ! [X: int,Xa: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa))
       => ~ ( ( Y = $ite(
                  ( member_int2(X,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                  & member_int2(Xa,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) ),
                  aa_int_int(uminus_uminus_int,
                    zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                      & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),
                  aa_int_int(aa_int_fun_int_int(plus_plus_int,
                      zero_n2684676970156552555ol_int(( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),X)
                        & ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),Xa) ))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,X),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,Xa),numeral_numeral_int(bit0(one)))))) ) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(X,Xa)) ) ) ) ).

% and_int.pelims
tff(fact_4578_mask__numeral,axiom,
    ! [N: num] : bit_se2119862282449309892nteger(numeral_numeral_nat(N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_se2119862282449309892nteger(pred_numeral(N)))) ).

% mask_numeral
tff(fact_4579_mask__numeral,axiom,
    ! [N: num] : bit_se943457434206027407atural(numeral_numeral_nat(N)) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se943457434206027407atural(pred_numeral(N)))) ).

% mask_numeral
tff(fact_4580_mask__numeral,axiom,
    ! [N: num] : aa_nat_int(bit_se2000444600071755411sk_int,numeral_numeral_nat(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,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,pred_numeral(N)))) ).

% mask_numeral
tff(fact_4581_mask__numeral,axiom,
    ! [N: num] : aa_nat_nat(bit_se2002935070580805687sk_nat,numeral_numeral_nat(N)) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,pred_numeral(N)))) ).

% mask_numeral
tff(fact_4582_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(numeral_numeral_int(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_4583_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% xor_numerals(4)
tff(fact_4584_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% xor_numerals(4)
tff(fact_4585_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit1(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% xor_numerals(4)
tff(fact_4586_xor__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit1(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% xor_numerals(4)
tff(fact_4587_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% xor_numerals(6)
tff(fact_4588_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% xor_numerals(6)
tff(fact_4589_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit0(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% xor_numerals(6)
tff(fact_4590_xor__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit0(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% xor_numerals(6)
tff(fact_4591_xor__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),one_one_Code_integer) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))) ).

% xor_one_eq
tff(fact_4592_xor__one__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(A,one_one_Code_natural) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))),zero_n8403883297036319079atural(~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))) ).

% xor_one_eq
tff(fact_4593_xor__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))) ).

% xor_one_eq
tff(fact_4594_xor__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_n2684676970156552555ol_int(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)))),zero_n2684676970156552555ol_int(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% xor_one_eq
tff(fact_4595_one__xor__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))),zero_n356916108424825756nteger(~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))) ).

% one_xor_eq
tff(fact_4596_one__xor__eq,axiom,
    ! [A: code_natural] : bit_se2046307713759805098atural(one_one_Code_natural,A) = minus_7197305767214868737atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))),zero_n8403883297036319079atural(~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))) ).

% one_xor_eq
tff(fact_4597_one__xor__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A)))),zero_n2687167440665602831ol_nat(~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))) ).

% one_xor_eq
tff(fact_4598_one__xor__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_n2684676970156552555ol_int(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A)))),zero_n2684676970156552555ol_int(~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% one_xor_eq
tff(fact_4599_mask__Suc__0,axiom,
    bit_se2119862282449309892nteger(aa_nat_nat(suc,zero_zero_nat)) = one_one_Code_integer ).

% mask_Suc_0
tff(fact_4600_mask__Suc__0,axiom,
    aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,zero_zero_nat)) = one_one_int ).

% mask_Suc_0
tff(fact_4601_mask__Suc__0,axiom,
    aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,zero_zero_nat)) = one_one_nat ).

% mask_Suc_0
tff(fact_4602_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : bit_se1745604003318907178nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = bit_se2119862282449309892nteger(N) ).

% take_bit_minus_one_eq_mask
tff(fact_4603_take__bit__minus__one__eq__mask,axiom,
    ! [N: nat] : aa_int_int(bit_se2923211474154528505it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_nat_int(bit_se2000444600071755411sk_int,N) ).

% take_bit_minus_one_eq_mask
tff(fact_4604_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% xor_numerals(3)
tff(fact_4605_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% xor_numerals(3)
tff(fact_4606_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% xor_numerals(3)
tff(fact_4607_xor__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit0(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% xor_numerals(3)
tff(fact_4608_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(Y))) = numera6620942414471956472nteger(bit1(Y)) ).

% xor_numerals(1)
tff(fact_4609_xor__numerals_I1_J,axiom,
    ! [Y: num] : bit_se2046307713759805098atural(one_one_Code_natural,numera5444537566228673987atural(bit0(Y))) = numera5444537566228673987atural(bit1(Y)) ).

% xor_numerals(1)
tff(fact_4610_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(bit1(Y)) ).

% xor_numerals(1)
tff(fact_4611_xor__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),numeral_numeral_int(bit0(Y))) = numeral_numeral_int(bit1(Y)) ).

% xor_numerals(1)
tff(fact_4612_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,one_one_Code_integer),numera6620942414471956472nteger(bit1(Y))) = numera6620942414471956472nteger(bit0(Y)) ).

% xor_numerals(2)
tff(fact_4613_xor__numerals_I2_J,axiom,
    ! [Y: num] : bit_se2046307713759805098atural(one_one_Code_natural,numera5444537566228673987atural(bit1(Y))) = numera5444537566228673987atural(bit0(Y)) ).

% xor_numerals(2)
tff(fact_4614_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,one_one_nat),numeral_numeral_nat(bit1(Y))) = numeral_numeral_nat(bit0(Y)) ).

% xor_numerals(2)
tff(fact_4615_xor__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,one_one_int),numeral_numeral_int(bit1(Y))) = numeral_numeral_int(bit0(Y)) ).

% xor_numerals(2)
tff(fact_4616_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit0(X))),one_one_Code_integer) = numera6620942414471956472nteger(bit1(X)) ).

% xor_numerals(5)
tff(fact_4617_xor__numerals_I5_J,axiom,
    ! [X: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit0(X)),one_one_Code_natural) = numera5444537566228673987atural(bit1(X)) ).

% xor_numerals(5)
tff(fact_4618_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit0(X))),one_one_nat) = numeral_numeral_nat(bit1(X)) ).

% xor_numerals(5)
tff(fact_4619_xor__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit0(X))),one_one_int) = numeral_numeral_int(bit1(X)) ).

% xor_numerals(5)
tff(fact_4620_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit1(X))),one_one_Code_integer) = numera6620942414471956472nteger(bit0(X)) ).

% xor_numerals(8)
tff(fact_4621_xor__numerals_I8_J,axiom,
    ! [X: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit1(X)),one_one_Code_natural) = numera5444537566228673987atural(bit0(X)) ).

% xor_numerals(8)
tff(fact_4622_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit1(X))),one_one_nat) = numeral_numeral_nat(bit0(X)) ).

% xor_numerals(8)
tff(fact_4623_xor__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit1(X))),one_one_int) = numeral_numeral_int(bit0(X)) ).

% xor_numerals(8)
tff(fact_4624_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% xor_numerals(7)
tff(fact_4625_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se2046307713759805098atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se2046307713759805098atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% xor_numerals(7)
tff(fact_4626_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit1(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% xor_numerals(7)
tff(fact_4627_xor__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit1(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% xor_numerals(7)
tff(fact_4628_flip__bit__nat__def,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M2),one_one_nat)) ).

% flip_bit_nat_def
tff(fact_4629_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_4630_flip__bit__eq__xor,axiom,
    ! [N: nat,A: code_integer] : bit_se1345352211410354436nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,A),bit_se7788150548672797655nteger(N,one_one_Code_integer)) ).

% flip_bit_eq_xor
tff(fact_4631_flip__bit__eq__xor,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se2159334234014336723it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% flip_bit_eq_xor
tff(fact_4632_flip__bit__eq__xor,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se2161824704523386999it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se6528837805403552850or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).

% flip_bit_eq_xor
tff(fact_4633_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_4634_mask__half__int,axiom,
    ! [N: nat] : aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_nat_int(bit_se2000444600071755411sk_int,N)),numeral_numeral_int(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_4635_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,numeral_numeral_nat(bit0(one))),N)),one_one_nat) ).

% mask_nat_def
tff(fact_4636_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(numeral_numeral_int(bit0(one))),N)),one_one_int) ).

% mask_int_def
tff(fact_4637_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : bit_se2119862282449309892nteger(N) = minus_8373710615458151222nteger(power_8256067586552552935nteger(numera6620942414471956472nteger(bit0(one)),N),one_one_Code_integer) ).

% mask_eq_exp_minus_1
tff(fact_4638_mask__eq__exp__minus__1,axiom,
    ! [N: nat] : bit_se943457434206027407atural(N) = minus_7197305767214868737atural(power_7079662738309270450atural(numera5444537566228673987atural(bit0(one)),N),one_one_Code_natural) ).

% mask_eq_exp_minus_1
tff(fact_4639_mask__eq__exp__minus__1,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(numeral_numeral_int(bit0(one))),N)),one_one_int) ).

% mask_eq_exp_minus_1
tff(fact_4640_mask__eq__exp__minus__1,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,numeral_numeral_nat(bit0(one))),N)),one_one_nat) ).

% mask_eq_exp_minus_1
tff(fact_4641_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(A0,A1))
     => ( ! [K2: int,L3: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(bit_and_int_rel),product_Pair_int_int(K2,L3))
           => ( ( ~ ( member_int2(K2,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int)))
                    & member_int2(L3,insert_int2(zero_zero_int,insert_int2(aa_int_int(uminus_uminus_int,one_one_int),bot_bot_set_int))) )
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(divide_divide_int,K2),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L3),numeral_numeral_int(bit0(one)))) )
             => aa_int_o(aa_int_fun_int_o(Pa,K2),L3) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4642_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I2,J))
     => ( aa_int_list_int(upto(I2),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J),cons_int(I2,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J)),nil_int) ) ) ).

% upto.psimps
tff(fact_4643_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( aa_int_list_int(upto(X),Xa) = Y )
     => ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa))
       => ~ ( ( Y = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X),Xa),cons_int(X,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,X),one_one_int)),Xa)),nil_int) )
           => ~ aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(X,Xa)) ) ) ) ).

% upto.pelims
tff(fact_4644_upto_Opinduct,axiom,
    ! [A0: int,A1: int,Pa: fun_int_fun_int_o] :
      ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(A0,A1))
     => ( ! [I3: int,J2: int] :
            ( aa_Pro4592068669816069595_int_o(accp_P1096762738010456898nt_int(upto_rel),product_Pair_int_int(I3,J2))
           => ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I3),J2)
               => aa_int_o(aa_int_fun_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,I3),one_one_int)),J2) )
             => aa_int_o(aa_int_fun_int_o(Pa,I3),J2) ) )
       => aa_int_o(aa_int_fun_int_o(Pa,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4645_nat0__intermed__int__val,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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(F,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I3),one_one_nat))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_4646_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% or_numerals(7)
tff(fact_4647_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se9127793120404214118atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% or_numerals(7)
tff(fact_4648_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit1(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% or_numerals(7)
tff(fact_4649_or__numerals_I7_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit1(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% or_numerals(7)
tff(fact_4650_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit1(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% or_numerals(6)
tff(fact_4651_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit1(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se9127793120404214118atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% or_numerals(6)
tff(fact_4652_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(X))),numeral_numeral_int(bit0(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% or_numerals(6)
tff(fact_4653_or__numerals_I6_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit1(X))),numeral_numeral_nat(bit0(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% or_numerals(6)
tff(fact_4654_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit1(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y)))) ).

% or_numerals(4)
tff(fact_4655_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit1(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se9127793120404214118atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y)))) ).

% or_numerals(4)
tff(fact_4656_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit1(Y))) = 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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y)))) ).

% or_numerals(4)
tff(fact_4657_or__numerals_I4_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit1(Y))) = 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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y)))) ).

% or_numerals(4)
tff(fact_4658_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit1(M2))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(M2)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N))))) ).

% and_not_numerals(8)
tff(fact_4659_abs__idempotent,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(abs_abs_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_idempotent
tff(fact_4660_abs__idempotent,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(abs_abs_Code_integer(A)) = abs_abs_Code_integer(A) ).

% abs_idempotent
tff(fact_4661_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int = aa_int_int(abs_abs_int,A) )
    <=> ( A = zero_zero_int ) ) ).

% abs_0_eq
tff(fact_4662_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat = abs_abs_rat(A) )
    <=> ( A = zero_zero_rat ) ) ).

% abs_0_eq
tff(fact_4663_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger = abs_abs_Code_integer(A) )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
tff(fact_4664_abs__eq__0,axiom,
    ! [A: int] :
      ( ( aa_int_int(abs_abs_int,A) = zero_zero_int )
    <=> ( A = zero_zero_int ) ) ).

% abs_eq_0
tff(fact_4665_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat(A) = zero_zero_rat )
    <=> ( A = zero_zero_rat ) ) ).

% abs_eq_0
tff(fact_4666_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer(A) = zero_z3403309356797280102nteger )
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
tff(fact_4667_abs__zero,axiom,
    aa_int_int(abs_abs_int,zero_zero_int) = zero_zero_int ).

% abs_zero
tff(fact_4668_abs__zero,axiom,
    abs_abs_rat(zero_zero_rat) = zero_zero_rat ).

% abs_zero
tff(fact_4669_abs__zero,axiom,
    abs_abs_Code_integer(zero_z3403309356797280102nteger) = zero_z3403309356797280102nteger ).

% abs_zero
tff(fact_4670_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_add_abs
tff(fact_4671_abs__add__abs,axiom,
    ! [A: rat,B: rat] : abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B)) ).

% abs_add_abs
tff(fact_4672_abs__add__abs,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_add_abs
tff(fact_4673_abs__mult__self__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,A)) = aa_int_int(aa_int_fun_int_int(times_times_int,A),A) ).

% abs_mult_self_eq
tff(fact_4674_abs__mult__self__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(A)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),A) ).

% abs_mult_self_eq
tff(fact_4675_abs__1,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_1
tff(fact_4676_abs__1,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_1
tff(fact_4677_abs__1,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_1
tff(fact_4678_abs__minus__cancel,axiom,
    ! [A: int] : aa_int_int(abs_abs_int,aa_int_int(uminus_uminus_int,A)) = aa_int_int(abs_abs_int,A) ).

% abs_minus_cancel
tff(fact_4679_abs__minus__cancel,axiom,
    ! [A: rat] : abs_abs_rat(uminus_uminus_rat(A)) = abs_abs_rat(A) ).

% abs_minus_cancel
tff(fact_4680_abs__minus__cancel,axiom,
    ! [A: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)) = abs_abs_Code_integer(A) ).

% abs_minus_cancel
tff(fact_4681_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_nonneg
tff(fact_4682_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_nonneg
tff(fact_4683_abs__of__nonneg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_nonneg
tff(fact_4684_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),A)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A) ) ).

% abs_le_self_iff
tff(fact_4685_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),A)
    <=> aa_rat_o(ord_less_eq_rat(zero_zero_rat),A) ) ).

% abs_le_self_iff
tff(fact_4686_abs__le__self__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),A)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A) ) ).

% abs_le_self_iff
tff(fact_4687_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),zero_z3403309356797280102nteger)
    <=> ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
tff(fact_4688_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),zero_zero_rat)
    <=> ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
tff(fact_4689_abs__le__zero__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),zero_zero_int)
    <=> ( A = zero_zero_int ) ) ).

% abs_le_zero_iff
tff(fact_4690_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),abs_abs_Code_integer(A))
    <=> ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
tff(fact_4691_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),abs_abs_rat(A))
    <=> ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
tff(fact_4692_zero__less__abs__iff,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_int_int(abs_abs_int,A))
    <=> ( A != zero_zero_int ) ) ).

% zero_less_abs_iff
tff(fact_4693_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_4694_abs__neg__one,axiom,
    abs_abs_rat(uminus_uminus_rat(one_one_rat)) = one_one_rat ).

% abs_neg_one
tff(fact_4695_abs__neg__one,axiom,
    abs_abs_Code_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = one_one_Code_integer ).

% abs_neg_one
tff(fact_4696_bit_Odisj__one__right,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_right
tff(fact_4697_bit_Odisj__one__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_right
tff(fact_4698_bit_Odisj__one__left,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_one_left
tff(fact_4699_bit_Odisj__one__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_one_left
tff(fact_4700_zdvd1__eq,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),one_one_int)
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% zdvd1_eq
tff(fact_4701_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_nonpos
tff(fact_4702_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_nonpos
tff(fact_4703_abs__of__nonpos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_nonpos
tff(fact_4704_bit_Ocompl__zero,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.compl_zero
tff(fact_4705_bit_Ocompl__zero,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.compl_zero
tff(fact_4706_bit_Ocompl__one,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = zero_z3403309356797280102nteger ).

% bit.compl_one
tff(fact_4707_bit_Ocompl__one,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(uminus_uminus_int,one_one_int)) = zero_zero_int ).

% bit.compl_one
tff(fact_4708_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),numera6620942414471956472nteger(bit1(Y))) = numera6620942414471956472nteger(bit1(Y)) ).

% or_numerals(2)
tff(fact_4709_or__numerals_I2_J,axiom,
    ! [Y: num] : bit_se9127793120404214118atural(one_one_Code_natural,numera5444537566228673987atural(bit1(Y))) = numera5444537566228673987atural(bit1(Y)) ).

% or_numerals(2)
tff(fact_4710_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),numeral_numeral_int(bit1(Y))) = numeral_numeral_int(bit1(Y)) ).

% or_numerals(2)
tff(fact_4711_or__numerals_I2_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),numeral_numeral_nat(bit1(Y))) = numeral_numeral_nat(bit1(Y)) ).

% or_numerals(2)
tff(fact_4712_or__numerals_I8_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit1(X))),one_one_Code_integer) = numera6620942414471956472nteger(bit1(X)) ).

% or_numerals(8)
tff(fact_4713_or__numerals_I8_J,axiom,
    ! [X: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit1(X)),one_one_Code_natural) = numera5444537566228673987atural(bit1(X)) ).

% or_numerals(8)
tff(fact_4714_or__numerals_I8_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(X))),one_one_int) = numeral_numeral_int(bit1(X)) ).

% or_numerals(8)
tff(fact_4715_or__numerals_I8_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit1(X))),one_one_nat) = numeral_numeral_nat(bit1(X)) ).

% or_numerals(8)
tff(fact_4716_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = one_one_int ) ) ).

% abs_sgn_eq_1
tff(fact_4717_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( abs_abs_rat(sgn_sgn_rat(A)) = one_one_rat ) ) ).

% abs_sgn_eq_1
tff(fact_4718_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
tff(fact_4719_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,numeral_numeral_int(bit1(N)))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(N))) ).

% or_minus_numerals(2)
tff(fact_4720_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,numeral_numeral_int(bit1(N)))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit1(N))) ).

% or_minus_numerals(6)
tff(fact_4721_bit_Odisj__cancel__right,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_cancel_right
tff(fact_4722_bit_Odisj__cancel__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_cancel_right
tff(fact_4723_bit_Odisj__cancel__left,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.disj_cancel_left
tff(fact_4724_bit_Odisj__cancel__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.disj_cancel_left
tff(fact_4725_bit_Oxor__cancel__right,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_right
tff(fact_4726_bit_Oxor__cancel__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(bit_ri7919022796975470100ot_int,X)) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_right
tff(fact_4727_bit_Oxor__cancel__left,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X)),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) ).

% bit.xor_cancel_left
tff(fact_4728_bit_Oxor__cancel__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(bit_ri7919022796975470100ot_int,X)),X) = aa_int_int(uminus_uminus_int,one_one_int) ).

% bit.xor_cancel_left
tff(fact_4729_bit_Oxor__one__right,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,X),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ).

% bit.xor_one_right
tff(fact_4730_bit_Oxor__one__right,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,X),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_right
tff(fact_4731_bit_Oxor__one__left,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3222712562003087583nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)),X) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) ).

% bit.xor_one_left
tff(fact_4732_bit_Oxor__one__left,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(bit_se6526347334894502574or_int,aa_int_int(uminus_uminus_int,one_one_int)),X) = aa_int_int(bit_ri7919022796975470100ot_int,X) ).

% bit.xor_one_left
tff(fact_4733_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_4734_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit0(X))),numera6620942414471956472nteger(bit0(Y))) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(X)),numera6620942414471956472nteger(Y))) ).

% or_numerals(3)
tff(fact_4735_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit0(X)),numera5444537566228673987atural(bit0(Y))) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se9127793120404214118atural(numera5444537566228673987atural(X),numera5444537566228673987atural(Y))) ).

% or_numerals(3)
tff(fact_4736_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit0(X))),numeral_numeral_int(bit0(Y))) = aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(X)),numeral_numeral_int(Y))) ).

% or_numerals(3)
tff(fact_4737_or__numerals_I3_J,axiom,
    ! [X: num,Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit0(X))),numeral_numeral_nat(bit0(Y))) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(X)),numeral_numeral_nat(Y))) ).

% or_numerals(3)
tff(fact_4738_or__numerals_I5_J,axiom,
    ! [X: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,numera6620942414471956472nteger(bit0(X))),one_one_Code_integer) = numera6620942414471956472nteger(bit1(X)) ).

% or_numerals(5)
tff(fact_4739_or__numerals_I5_J,axiom,
    ! [X: num] : bit_se9127793120404214118atural(numera5444537566228673987atural(bit0(X)),one_one_Code_natural) = numera5444537566228673987atural(bit1(X)) ).

% or_numerals(5)
tff(fact_4740_or__numerals_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit0(X))),one_one_int) = numeral_numeral_int(bit1(X)) ).

% or_numerals(5)
tff(fact_4741_or__numerals_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,numeral_numeral_nat(bit0(X))),one_one_nat) = numeral_numeral_nat(bit1(X)) ).

% or_numerals(5)
tff(fact_4742_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),numera6620942414471956472nteger(bit0(Y))) = numera6620942414471956472nteger(bit1(Y)) ).

% or_numerals(1)
tff(fact_4743_or__numerals_I1_J,axiom,
    ! [Y: num] : bit_se9127793120404214118atural(one_one_Code_natural,numera5444537566228673987atural(bit0(Y))) = numera5444537566228673987atural(bit1(Y)) ).

% or_numerals(1)
tff(fact_4744_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),numeral_numeral_int(bit0(Y))) = numeral_numeral_int(bit1(Y)) ).

% or_numerals(1)
tff(fact_4745_or__numerals_I1_J,axiom,
    ! [Y: num] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),numeral_numeral_nat(bit0(Y))) = numeral_numeral_nat(bit1(Y)) ).

% or_numerals(1)
tff(fact_4746_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : bit_se7788150548672797655nteger(N,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,bit_se2119862282449309892nteger(N)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_4747_push__bit__minus__one__eq__not__mask,axiom,
    ! [N: nat] : aa_int_int(bit_se545348938243370406it_int(N),aa_int_int(uminus_uminus_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,aa_nat_int(bit_se2000444600071755411sk_int,N)) ).

% push_bit_minus_one_eq_not_mask
tff(fact_4748_not__one__eq,axiom,
    aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,one_one_Code_integer) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,numera6620942414471956472nteger(bit0(one))) ).

% not_one_eq
tff(fact_4749_not__one__eq,axiom,
    aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit0(one))) ).

% not_one_eq
tff(fact_4750_or__minus__minus__numerals,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_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,numeral_numeral_int(M2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(N)),one_one_int))) ).

% or_minus_minus_numerals
tff(fact_4751_and__minus__minus__numerals,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(uminus_uminus_int,numeral_numeral_int(M2))),aa_int_int(uminus_uminus_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,numeral_numeral_int(M2)),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(N)),one_one_int))) ).

% and_minus_minus_numerals
tff(fact_4752_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_4753_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(A),B) ) ).

% abs_le_D1
tff(fact_4754_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
     => aa_rat_o(ord_less_eq_rat(A),B) ) ).

% abs_le_D1
tff(fact_4755_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ).

% abs_le_D1
tff(fact_4756_abs__ge__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(A),abs_abs_Code_integer(A)) ).

% abs_ge_self
tff(fact_4757_abs__ge__self,axiom,
    ! [A: rat] : aa_rat_o(ord_less_eq_rat(A),abs_abs_rat(A)) ).

% abs_ge_self
tff(fact_4758_abs__ge__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(abs_abs_int,A)) ).

% abs_ge_self
tff(fact_4759_abs__mult,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ).

% abs_mult
tff(fact_4760_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ).

% abs_mult
tff(fact_4761_abs__one,axiom,
    abs_abs_Code_integer(one_one_Code_integer) = one_one_Code_integer ).

% abs_one
tff(fact_4762_abs__one,axiom,
    abs_abs_rat(one_one_rat) = one_one_rat ).

% abs_one
tff(fact_4763_abs__one,axiom,
    aa_int_int(abs_abs_int,one_one_int) = one_one_int ).

% abs_one
tff(fact_4764_abs__minus__commute,axiom,
    ! [A: int,B: int] : aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B)) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A)) ).

% abs_minus_commute
tff(fact_4765_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] : abs_abs_Code_integer(minus_8373710615458151222nteger(A,B)) = abs_abs_Code_integer(minus_8373710615458151222nteger(B,A)) ).

% abs_minus_commute
tff(fact_4766_set__bit__nat__def,axiom,
    ! [M2: nat,N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,M2),N) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,M2),one_one_nat)) ).

% set_bit_nat_def
tff(fact_4767_or__not__numerals_I4_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit0(M2))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,one_one_int) ).

% or_not_numerals(4)
tff(fact_4768_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,numeral_numeral_int(bit0(N)))) = aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N))) ).

% or_not_numerals(2)
tff(fact_4769_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,numeral_numeral_int(bit1(N)))) = aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(bit0(N))) ).

% or_not_numerals(3)
tff(fact_4770_or__not__numerals_I7_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(M2))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = aa_int_int(bit_ri7919022796975470100ot_int,zero_zero_int) ).

% or_not_numerals(7)
tff(fact_4771_bit_Ocompl__unique,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,X),Y) = zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,X),Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,X) = Y ) ) ) ).

% bit.compl_unique
tff(fact_4772_bit_Ocompl__unique,axiom,
    ! [X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,X),Y) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,X),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( aa_int_int(bit_ri7919022796975470100ot_int,X) = Y ) ) ) ).

% bit.compl_unique
tff(fact_4773_abs__ge__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),abs_abs_Code_integer(A)) ).

% abs_ge_zero
tff(fact_4774_abs__ge__zero,axiom,
    ! [A: rat] : aa_rat_o(ord_less_eq_rat(zero_zero_rat),abs_abs_rat(A)) ).

% abs_ge_zero
tff(fact_4775_abs__ge__zero,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),aa_int_int(abs_abs_int,A)) ).

% abs_ge_zero
tff(fact_4776_abs__not__less__zero,axiom,
    ! [A: code_integer] : ~ aa_Code_integer_o(ord_le6747313008572928689nteger(abs_abs_Code_integer(A)),zero_z3403309356797280102nteger) ).

% abs_not_less_zero
tff(fact_4777_abs__not__less__zero,axiom,
    ! [A: rat] : ~ aa_rat_o(ord_less_rat(abs_abs_rat(A)),zero_zero_rat) ).

% abs_not_less_zero
tff(fact_4778_abs__not__less__zero,axiom,
    ! [A: int] : ~ aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),zero_zero_int) ).

% abs_not_less_zero
tff(fact_4779_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),A)
     => ( abs_abs_Code_integer(A) = A ) ) ).

% abs_of_pos
tff(fact_4780_abs__of__pos,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(zero_zero_rat),A)
     => ( abs_abs_rat(A) = A ) ) ).

% abs_of_pos
tff(fact_4781_abs__of__pos,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),A)
     => ( aa_int_int(abs_abs_int,A) = A ) ) ).

% abs_of_pos
tff(fact_4782_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq
tff(fact_4783_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq
tff(fact_4784_abs__triangle__ineq,axiom,
    ! [A: int,B: 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,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq
tff(fact_4785_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,A))) ).

% abs_triangle_ineq2_sym
tff(fact_4786_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,abs_abs_rat(A)),abs_abs_rat(B))),abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_4787_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: 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,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),A))) ).

% abs_triangle_ineq2_sym
tff(fact_4788_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B)))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq3
tff(fact_4789_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,abs_abs_rat(A)),abs_abs_rat(B)))),abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq3
tff(fact_4790_abs__triangle__ineq3,axiom,
    ! [A: int,B: 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,A)),aa_int_int(abs_abs_int,B)))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq3
tff(fact_4791_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(minus_8373710615458151222nteger(abs_abs_Code_integer(A),abs_abs_Code_integer(B))),abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))) ).

% abs_triangle_ineq2
tff(fact_4792_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,abs_abs_rat(A)),abs_abs_rat(B))),abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))) ).

% abs_triangle_ineq2
tff(fact_4793_abs__triangle__ineq2,axiom,
    ! [A: int,B: 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,A)),aa_int_int(abs_abs_int,B))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),B))) ).

% abs_triangle_ineq2
tff(fact_4794_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D2: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(abs_abs_Code_integer(A)),C)
     => ( aa_Code_integer_o(ord_le6747313008572928689nteger(abs_abs_Code_integer(B)),D2)
       => aa_Code_integer_o(ord_le6747313008572928689nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,C),D2)) ) ) ).

% abs_mult_less
tff(fact_4795_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D2: rat] :
      ( aa_rat_o(ord_less_rat(abs_abs_rat(A)),C)
     => ( aa_rat_o(ord_less_rat(abs_abs_rat(B)),D2)
       => aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B))),aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,C),D2)) ) ) ).

% abs_mult_less
tff(fact_4796_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D2: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,A)),C)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,B)),D2)
       => 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,A)),aa_int_int(abs_abs_int,B))),aa_int_int(aa_int_fun_int_int(times_times_int,C),D2)) ) ) ).

% abs_mult_less
tff(fact_4797_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
     => ( aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B)
       => aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),B) ) ) ).

% abs_leI
tff(fact_4798_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),B)
       => aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B) ) ) ).

% abs_leI
tff(fact_4799_abs__leI,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B) ) ) ).

% abs_leI
tff(fact_4800_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),B)
     => aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ).

% abs_le_D2
tff(fact_4801_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
     => aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),B) ) ).

% abs_le_D2
tff(fact_4802_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ).

% abs_le_D2
tff(fact_4803_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(A)),B)
    <=> ( aa_Code_integer_o(ord_le3102999989581377725nteger(A),B)
        & aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),B) ) ) ).

% abs_le_iff
tff(fact_4804_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(abs_abs_rat(A)),B)
    <=> ( aa_rat_o(ord_less_eq_rat(A),B)
        & aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),B) ) ) ).

% abs_le_iff
tff(fact_4805_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,A)),B)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),B) ) ) ).

% abs_le_iff
tff(fact_4806_abs__ge__minus__self,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A)),abs_abs_Code_integer(A)) ).

% abs_ge_minus_self
tff(fact_4807_abs__ge__minus__self,axiom,
    ! [A: rat] : aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(A)),abs_abs_rat(A)) ).

% abs_ge_minus_self
tff(fact_4808_abs__ge__minus__self,axiom,
    ! [A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(uminus_uminus_int,A)),aa_int_int(abs_abs_int,A)) ).

% abs_ge_minus_self
tff(fact_4809_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: rat] : abs_abs_rat(K) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,K),sgn_sgn_rat(K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4810_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,K) = aa_int_int(aa_int_fun_int_int(times_times_int,K),aa_int_int(sgn_sgn_int,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4811_linordered__idom__class_Oabs__sgn,axiom,
    ! [K: code_integer] : abs_abs_Code_integer(K) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,K),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,K)) ).

% linordered_idom_class.abs_sgn
tff(fact_4812_abs__mult__sgn,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),sgn_sgn_rat(A)) = A ).

% abs_mult_sgn
tff(fact_4813_abs__mult__sgn,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(sgn_sgn_int,A)) = A ).

% abs_mult_sgn
tff(fact_4814_abs__mult__sgn,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = A ).

% abs_mult_sgn
tff(fact_4815_sgn__mult__abs,axiom,
    ! [A: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(A)),abs_abs_rat(A)) = A ).

% sgn_mult_abs
tff(fact_4816_sgn__mult__abs,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,A)),aa_int_int(abs_abs_int,A)) = A ).

% sgn_mult_abs
tff(fact_4817_sgn__mult__abs,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)),abs_abs_Code_integer(A)) = A ).

% sgn_mult_abs
tff(fact_4818_mult__sgn__abs,axiom,
    ! [X: rat] : aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,sgn_sgn_rat(X)),abs_abs_rat(X)) = X ).

% mult_sgn_abs
tff(fact_4819_mult__sgn__abs,axiom,
    ! [X: int] : aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(sgn_sgn_int,X)),aa_int_int(abs_abs_int,X)) = X ).

% mult_sgn_abs
tff(fact_4820_mult__sgn__abs,axiom,
    ! [X: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,X)),abs_abs_Code_integer(X)) = X ).

% mult_sgn_abs
tff(fact_4821_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,M2),N)) = one_one_int )
     => ( aa_int_int(abs_abs_int,M2) = one_one_int ) ) ).

% abs_zmult_eq_1
tff(fact_4822_set__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)) ).

% set_bit_int_def
tff(fact_4823_bit_Oabstract__boolean__algebra__axioms,axiom,
    aa_Code_integer_o(boolea5643492148547655015nteger(bit_se3949692690581998587nteger,bit_se1080825931792720795nteger,bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer)) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_4824_bit_Oabstract__boolean__algebra__axioms,axiom,
    aa_int_o(aa_int_fun_int_o(boolea5469654029284665142ra_int(bit_se725231765392027082nd_int,bit_se1409905431419307370or_int,bit_ri7919022796975470100ot_int),zero_zero_int),aa_int_int(uminus_uminus_int,one_one_int)) ).

% bit.abstract_boolean_algebra_axioms
tff(fact_4825_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit0(M2))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(M2)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N))))) ).

% or_not_numerals(5)
tff(fact_4826_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E: rat] :
          ( aa_rat_o(ord_less_rat(zero_zero_rat),E)
         => aa_rat_o(ord_less_eq_rat(abs_abs_rat(X)),E) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
tff(fact_4827_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),X)
     => ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(Y)),X) = abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4828_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),X)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(Y)),X) = abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4829_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),X)
     => ( aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,Y)),X) = aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,Y),X)) ) ) ).

% abs_mult_pos
tff(fact_4830_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),A)
          | aa_Code_integer_o(ord_le3102999989581377725nteger(A),zero_z3403309356797280102nteger) )
        & ( aa_Code_integer_o(ord_le3102999989581377725nteger(zero_z3403309356797280102nteger),B)
          | aa_Code_integer_o(ord_le3102999989581377725nteger(B),zero_z3403309356797280102nteger) ) )
     => ( abs_abs_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,A),B)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B)) ) ) ).

% abs_eq_mult
tff(fact_4831_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),A)
          | aa_rat_o(ord_less_eq_rat(A),zero_zero_rat) )
        & ( aa_rat_o(ord_less_eq_rat(zero_zero_rat),B)
          | aa_rat_o(ord_less_eq_rat(B),zero_zero_rat) ) )
     => ( abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,A),B)) = aa_rat_rat(aa_rat_fun_rat_rat(times_times_rat,abs_abs_rat(A)),abs_abs_rat(B)) ) ) ).

% abs_eq_mult
tff(fact_4832_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),A)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),zero_zero_int) )
        & ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),B)
          | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),zero_zero_int) ) )
     => ( aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(times_times_int,A),B)) = aa_int_int(aa_int_fun_int_int(times_times_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B)) ) ) ).

% abs_eq_mult
tff(fact_4833_abs__minus__le__zero,axiom,
    ! [A: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,abs_abs_Code_integer(A))),zero_z3403309356797280102nteger) ).

% abs_minus_le_zero
tff(fact_4834_abs__minus__le__zero,axiom,
    ! [A: rat] : aa_rat_o(ord_less_eq_rat(uminus_uminus_rat(abs_abs_rat(A))),zero_zero_rat) ).

% abs_minus_le_zero
tff(fact_4835_abs__minus__le__zero,axiom,
    ! [A: 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,A))),zero_zero_int) ).

% abs_minus_le_zero
tff(fact_4836_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(A,B))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(A)),abs_abs_Code_integer(B))) ).

% abs_triangle_ineq4
tff(fact_4837_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),B))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(A)),abs_abs_rat(B))) ).

% abs_triangle_ineq4
tff(fact_4838_abs__triangle__ineq4,axiom,
    ! [A: int,B: 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,A),B))),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(abs_abs_int,A)),aa_int_int(abs_abs_int,B))) ).

% abs_triangle_ineq4
tff(fact_4839_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D2: code_integer] : aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),B),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,C),D2)))),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,abs_abs_Code_integer(minus_8373710615458151222nteger(A,C))),abs_abs_Code_integer(minus_8373710615458151222nteger(B,D2)))) ).

% abs_diff_triangle_ineq
tff(fact_4840_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D2: rat] : aa_rat_o(ord_less_eq_rat(abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,A),B)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,C),D2)))),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,A),C))),abs_abs_rat(aa_rat_rat(aa_rat_fun_rat_rat(minus_minus_rat,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_4841_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D2: 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,A),B)),aa_int_int(aa_int_fun_int_int(plus_plus_int,C),D2)))),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,A),C))),aa_int_int(abs_abs_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,B),D2)))) ).

% abs_diff_triangle_ineq
tff(fact_4842_abs__of__neg,axiom,
    ! [A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),zero_zero_int)
     => ( aa_int_int(abs_abs_int,A) = aa_int_int(uminus_uminus_int,A) ) ) ).

% abs_of_neg
tff(fact_4843_abs__of__neg,axiom,
    ! [A: rat] :
      ( aa_rat_o(ord_less_rat(A),zero_zero_rat)
     => ( abs_abs_rat(A) = uminus_uminus_rat(A) ) ) ).

% abs_of_neg
tff(fact_4844_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),zero_z3403309356797280102nteger)
     => ( abs_abs_Code_integer(A) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) ) ) ).

% abs_of_neg
tff(fact_4845_abs__sgn__eq,axiom,
    ! [A: int] :
      aa_int_int(abs_abs_int,aa_int_int(sgn_sgn_int,A)) = $ite(A = zero_zero_int,zero_zero_int,one_one_int) ).

% abs_sgn_eq
tff(fact_4846_abs__sgn__eq,axiom,
    ! [A: rat] :
      abs_abs_rat(sgn_sgn_rat(A)) = $ite(A = zero_zero_rat,zero_zero_rat,one_one_rat) ).

% abs_sgn_eq
tff(fact_4847_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      abs_abs_Code_integer(aa_Cod1783974652533396728nteger(sgn_sgn_Code_integer,A)) = $ite(A = zero_z3403309356797280102nteger,zero_z3403309356797280102nteger,one_one_Code_integer) ).

% abs_sgn_eq
tff(fact_4848_minus__eq__not__plus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A)),one_one_Code_integer) ).

% minus_eq_not_plus_1
tff(fact_4849_minus__eq__not__plus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(bit_ri7919022796975470100ot_int,A)),one_one_int) ).

% minus_eq_not_plus_1
tff(fact_4850_minus__eq__not__minus__1,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A) = aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,minus_8373710615458151222nteger(A,one_one_Code_integer)) ).

% minus_eq_not_minus_1
tff(fact_4851_minus__eq__not__minus__1,axiom,
    ! [A: int] : aa_int_int(uminus_uminus_int,A) = aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,A),one_one_int)) ).

% minus_eq_not_minus_1
tff(fact_4852_not__eq__complement,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,A) = minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,A),one_one_Code_integer) ).

% not_eq_complement
tff(fact_4853_not__eq__complement,axiom,
    ! [A: int] : aa_int_int(bit_ri7919022796975470100ot_int,A) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(uminus_uminus_int,A)),one_one_int) ).

% not_eq_complement
tff(fact_4854_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_4855_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_4856_set__bit__eq__or,axiom,
    ! [N: nat,A: code_integer] : bit_se2793503036327961859nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),bit_se7788150548672797655nteger(N,one_one_Code_integer)) ).

% set_bit_eq_or
tff(fact_4857_set__bit__eq__or,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se7879613467334960850it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),aa_int_int(bit_se545348938243370406it_int(N),one_one_int)) ).

% set_bit_eq_or
tff(fact_4858_set__bit__eq__or,axiom,
    ! [N: nat,A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se7882103937844011126it_nat,N),A) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(bit_se547839408752420682it_nat,N),one_one_nat)) ).

% set_bit_eq_or
tff(fact_4859_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(M2))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(M2)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N))))) ).

% or_not_numerals(8)
tff(fact_4860_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(bit1(M2))),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(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,numeral_numeral_int(bit0(one))),aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,numeral_numeral_int(M2)),aa_int_int(bit_ri7919022796975470100ot_int,numeral_numeral_int(N))))) ).

% or_not_numerals(9)
tff(fact_4861_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_4862_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,one_one_Code_integer),abs_abs_Code_integer(X))) ).

% abs_add_one_gt_zero
tff(fact_4863_abs__add__one__gt__zero,axiom,
    ! [X: rat] : aa_rat_o(ord_less_rat(zero_zero_rat),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,one_one_rat),abs_abs_rat(X))) ).

% abs_add_one_gt_zero
tff(fact_4864_abs__add__one__gt__zero,axiom,
    ! [X: 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,X))) ).

% abs_add_one_gt_zero
tff(fact_4865_bit_Ocomplement__unique,axiom,
    ! [A: code_integer,X: code_integer,Y: code_integer] :
      ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),X) = zero_z3403309356797280102nteger )
     => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),X) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
       => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),Y) = zero_z3403309356797280102nteger )
         => ( ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),Y) = aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4866_bit_Ocomplement__unique,axiom,
    ! [A: int,X: int,Y: int] :
      ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),X) = zero_zero_int )
     => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),X) = aa_int_int(uminus_uminus_int,one_one_int) )
       => ( ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),Y) = zero_zero_int )
         => ( ( aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),Y) = aa_int_int(uminus_uminus_int,one_one_int) )
           => ( X = Y ) ) ) ) ) ).

% bit.complement_unique
tff(fact_4867_and__not__numerals_I4_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit0(M2))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = numeral_numeral_int(bit0(M2)) ).

% and_not_numerals(4)
tff(fact_4868_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,numeral_numeral_int(bit0(N)))) = one_one_int ).

% and_not_numerals(2)
tff(fact_4869_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != 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,M2),N)),M2)
      <=> ( aa_int_int(abs_abs_int,N) = one_one_int ) ) ) ).

% zdvd_mult_cancel1
tff(fact_4870_unset__bit__eq__and__not,axiom,
    ! [N: nat,A: code_integer] : bit_se8260200283734997820nteger(N,A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,A),aa_Cod1783974652533396728nteger(bit_ri7632146776885996613nteger,bit_se7788150548672797655nteger(N,one_one_Code_integer))) ).

% unset_bit_eq_and_not
tff(fact_4871_unset__bit__eq__and__not,axiom,
    ! [N: nat,A: int] : aa_int_int(bit_se4203085406695923979it_int(N),A) = aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,A),aa_int_int(bit_ri7919022796975470100ot_int,aa_int_int(bit_se545348938243370406it_int(N),one_one_int))) ).

% unset_bit_eq_and_not
tff(fact_4872_abs__square__eq__1,axiom,
    ! [X: code_integer] :
      ( ( power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one))) = one_one_Code_integer )
    <=> ( abs_abs_Code_integer(X) = one_one_Code_integer ) ) ).

% abs_square_eq_1
tff(fact_4873_abs__square__eq__1,axiom,
    ! [X: rat] :
      ( ( power_power_rat(X,numeral_numeral_nat(bit0(one))) = one_one_rat )
    <=> ( abs_abs_rat(X) = one_one_rat ) ) ).

% abs_square_eq_1
tff(fact_4874_abs__square__eq__1,axiom,
    ! [X: int] :
      ( ( aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(one))) = one_one_int )
    <=> ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ).

% abs_square_eq_1
tff(fact_4875_and__not__numerals_I7_J,axiom,
    ! [M2: num] : aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,numeral_numeral_int(bit1(M2))),aa_int_int(bit_ri7919022796975470100ot_int,one_one_int)) = numeral_numeral_int(bit0(M2)) ).

% and_not_numerals(7)
tff(fact_4876_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,numeral_numeral_int(bit1(N)))) = zero_zero_int ).

% and_not_numerals(3)
tff(fact_4877_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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),numeral_numeral_int(bit0(one)))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(bit0(one))))))) ) ) ) ).

% xor_int_unfold
tff(fact_4878_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(ord_le3102999989581377725nteger(power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(ord_le3102999989581377725nteger(abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_le_1
tff(fact_4879_abs__square__le__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(ord_less_eq_rat(power_power_rat(X,numeral_numeral_nat(bit0(one)))),one_one_rat)
    <=> aa_rat_o(ord_less_eq_rat(abs_abs_rat(X)),one_one_rat) ) ).

% abs_square_le_1
tff(fact_4880_abs__square__le__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_le_1
tff(fact_4881_abs__square__less__1,axiom,
    ! [X: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(power_8256067586552552935nteger(X,numeral_numeral_nat(bit0(one)))),one_one_Code_integer)
    <=> aa_Code_integer_o(ord_le6747313008572928689nteger(abs_abs_Code_integer(X)),one_one_Code_integer) ) ).

% abs_square_less_1
tff(fact_4882_abs__square__less__1,axiom,
    ! [X: rat] :
      ( aa_rat_o(ord_less_rat(power_power_rat(X,numeral_numeral_nat(bit0(one)))),one_one_rat)
    <=> aa_rat_o(ord_less_rat(abs_abs_rat(X)),one_one_rat) ) ).

% abs_square_less_1
tff(fact_4883_abs__square__less__1,axiom,
    ! [X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_nat_int(power_power_int(X),numeral_numeral_nat(bit0(one)))),one_one_int)
    <=> aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(abs_abs_int,X)),one_one_int) ) ).

% abs_square_less_1
tff(fact_4884_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),I3)
            & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,M2)),K)
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
           => ? [I3: nat] :
                ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),I3)
                & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N)
                & ( aa_nat_int(F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_4885_decr__lemma,axiom,
    ! [D2: int,X: int,Z: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),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,X),Z))),one_one_int)),D2))),Z) ) ).

% decr_lemma
tff(fact_4886_incr__lemma,axiom,
    ! [D2: int,Z: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,Z),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),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,X),Z))),one_one_int)),D2))) ) ).

% incr_lemma
tff(fact_4887_or__one__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,A),one_one_Code_integer) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))) ).

% or_one_eq
tff(fact_4888_or__one__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(A,one_one_Code_natural) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))) ).

% or_one_eq
tff(fact_4889_or__one__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,A),one_one_int) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_n2684676970156552555ol_int(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% or_one_eq
tff(fact_4890_or__one__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,A),one_one_nat) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))) ).

% or_one_eq
tff(fact_4891_one__or__eq,axiom,
    ! [A: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),A) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),zero_n356916108424825756nteger(aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A))) ).

% one_or_eq
tff(fact_4892_one__or__eq,axiom,
    ! [A: code_natural] : bit_se9127793120404214118atural(one_one_Code_natural,A) = aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),zero_n8403883297036319079atural(aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A))) ).

% one_or_eq
tff(fact_4893_one__or__eq,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),A) = aa_int_int(aa_int_fun_int_int(plus_plus_int,A),zero_n2684676970156552555ol_int(aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A))) ).

% one_or_eq
tff(fact_4894_one__or__eq,axiom,
    ! [A: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),A) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),zero_n2687167440665602831ol_nat(aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A))) ).

% one_or_eq
tff(fact_4895_mask__Suc__double,axiom,
    ! [N: nat] : bit_se2119862282449309892nteger(aa_nat_nat(suc,N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se1080825931792720795nteger,one_one_Code_integer),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),bit_se2119862282449309892nteger(N))) ).

% mask_Suc_double
tff(fact_4896_mask__Suc__double,axiom,
    ! [N: nat] : bit_se943457434206027407atural(aa_nat_nat(suc,N)) = bit_se9127793120404214118atural(one_one_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),bit_se943457434206027407atural(N))) ).

% mask_Suc_double
tff(fact_4897_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_int(bit_se2000444600071755411sk_int,aa_nat_nat(suc,N)) = aa_int_int(aa_int_fun_int_int(bit_se1409905431419307370or_int,one_one_int),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),aa_nat_int(bit_se2000444600071755411sk_int,N))) ).

% mask_Suc_double
tff(fact_4898_mask__Suc__double,axiom,
    ! [N: nat] : aa_nat_nat(bit_se2002935070580805687sk_nat,aa_nat_nat(suc,N)) = aa_nat_nat(aa_nat_fun_nat_nat(bit_se1412395901928357646or_nat,one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(bit_se2002935070580805687sk_nat,N))) ).

% mask_Suc_double
tff(fact_4899_nat__ivt__aux,axiom,
    ! [N: nat,F: fun_nat_int,K: int] :
      ( ! [I3: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I3),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(F,aa_nat_nat(suc,I3))),aa_nat_int(F,I3)))),one_one_int) )
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_nat_int(F,zero_zero_nat)),K)
       => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,K),aa_nat_int(F,N))
         => ? [I3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I3),N)
              & ( aa_nat_int(F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_4900_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea2445317508997433345nteger(bit_se3949692690581998587nteger,bit_se1080825931792720795nteger,bit_ri7632146776885996613nteger,zero_z3403309356797280102nteger,aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),bit_se3222712562003087583nteger) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_4901_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    boolea8527374999097803216ff_int(bit_se725231765392027082nd_int,bit_se1409905431419307370or_int,bit_ri7919022796975470100ot_int,zero_zero_int,aa_int_int(uminus_uminus_int,one_one_int),bit_se6526347334894502574or_int) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_4902_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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(modulo_modulo_int,L),numeral_numeral_int(bit0(one))))),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(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),numeral_numeral_int(bit0(one)))),aa_int_int(aa_int_fun_int_int(divide_divide_int,L),numeral_numeral_int(bit0(one))))))) ) ) ).

% or_int_unfold
tff(fact_4903_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,numeral_numeral_int(bit0(N)))),one_one_int) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit_or_not_num_neg(one,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_4904_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,numeral_numeral_int(bit0(N)))) = aa_int_int(uminus_uminus_int,numeral_numeral_int(bit_or_not_num_neg(one,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_4905_product__nth,axiom,
    ! [N: nat,Xs: list_h2745205591237434579t_unit,Ys: list_set_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_s7011499401410611007t_unit(Xs)),size_s3254054031482475050et_nat(Ys)))
     => ( nth_Pr7683458668995092411et_nat(produc7481864959766642638et_nat(Xs,Ys),N) = produc7507926704131184380et_nat(nth_he7452997726511547988t_unit(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s3254054031482475050et_nat(Ys))),aa_nat_set_nat(nth_set_nat(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s3254054031482475050et_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4906_product__nth,axiom,
    ! [N: nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P9062070895058802706et_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_s7422308124397347553_nat_o(Xs)),size_s2864271983995610790et_nat(Ys)))
     => ( nth_Pr3203592224402014297et_nat(produc5339406111829240172et_nat(Xs,Ys),N) = produc6017135503889078682et_nat(nth_fu1272478342191624950_nat_o(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s2864271983995610790et_nat(Ys))),nth_Pr7683458668995092411et_nat(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s2864271983995610790et_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4907_product__nth,axiom,
    ! [N: nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P8836981409019564976et_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_s7422308124397347553_nat_o(Xs)),size_s6978193692292872260et_nat(Ys)))
     => ( nth_Pr5532225396715843703et_nat(produc5800657659535792522et_nat(Xs,Ys),N) = produc7291855603632349880et_nat(nth_fu1272478342191624950_nat_o(Xs,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_s6978193692292872260et_nat(Ys))),nth_Pr3203592224402014297et_nat(Ys,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_s6978193692292872260et_nat(Ys)))) ) ) ).

% product_nth
tff(fact_4908_product__nth,axiom,
    ! [N: nat,Xs: list_int,Ys: list_int] :
      ( 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_int(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr4439495888332055232nt_int(product_int_int(Xs,Ys),N) = product_Pair_int_int(aa_nat_int(nth_int(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4909_product__nth,axiom,
    ! [N: nat,Xs: list_int,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_size_list_int(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr8617346907841251940nt_nat(product_int_nat(Xs,Ys),N) = product_Pair_int_nat(aa_nat_int(nth_int(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_4910_product__nth,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_int] :
      ( 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_nat(Xs)),size_size_list_int(Ys)))
     => ( nth_Pr3440142176431000676at_int(product_nat_int(Xs,Ys),N) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,N),size_size_list_int(Ys))),aa_nat_int(nth_int(Ys),aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,N),size_size_list_int(Ys)))) ) ) ).

% product_nth
tff(fact_4911_product__nth,axiom,
    ! [N: nat,Xs: list_nat,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_size_list_nat(Xs)),size_size_list_nat(Ys)))
     => ( nth_Pr7617993195940197384at_nat(product_nat_nat(Xs,Ys),N) = product_Pair_nat_nat(aa_nat_nat(nth_nat(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_4912_drop__bit__rec,axiom,
    ! [N: nat,A: code_integer] :
      bit_se3928097537394005634nteger(N,A) = $ite(N = zero_zero_nat,A,bit_se3928097537394005634nteger(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one))))) ).

% drop_bit_rec
tff(fact_4913_drop__bit__rec,axiom,
    ! [N: nat,A: nat] :
      aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A) = $ite(N = zero_zero_nat,A,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one))))) ).

% drop_bit_rec
tff(fact_4914_drop__bit__rec,axiom,
    ! [N: nat,A: code_natural] :
      bit_se2751692689150723149atural(N,A) = $ite(N = zero_zero_nat,A,bit_se2751692689150723149atural(aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat),divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one))))) ).

% drop_bit_rec
tff(fact_4915_drop__bit__rec,axiom,
    ! [N: nat,A: int] :
      aa_int_int(bit_se8568078237143864401it_int(N),A) = $ite(N = zero_zero_nat,A,aa_int_int(bit_se8568078237143864401it_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,A),numeral_numeral_int(bit0(one))))) ).

% drop_bit_rec
tff(fact_4916_bit__rec,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
    <=> $ite(N = zero_zero_nat,~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A),aa_nat_o(bit_se9216721137139052372nteger(divide6298287555418463151nteger(A,numera6620942414471956472nteger(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_4917_bit__rec,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
    <=> $ite(N = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,A),numeral_numeral_nat(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_4918_bit__rec,axiom,
    ! [A: code_natural,N: nat] :
      ( aa_nat_o(bit_se8040316288895769887atural(A),N)
    <=> $ite(N = zero_zero_nat,~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A),aa_nat_o(bit_se8040316288895769887atural(divide5121882707175180666atural(A,numera5444537566228673987atural(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_4919_bit__rec,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
    <=> $ite(N = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),numeral_numeral_int(bit0(one)))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ) ).

% bit_rec
tff(fact_4920_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,numeral_numeral_nat(bit0(one))),A),aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4921_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,A),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_Code_integer_o(dvd_dvd_Code_integer(numera6620942414471956472nteger(bit0(one))),A),aa_nat_o(bit_se9216721137139052372nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4922_bit__sum__mult__2__cases,axiom,
    ! [A: code_natural,B: code_natural,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(bit_se8040316288895769887atural(A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,A),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_Code_natural_o(dvd_dvd_Code_natural(numera5444537566228673987atural(bit0(one))),A),aa_nat_o(bit_se8040316288895769887atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,numera5444537566228673987atural(bit0(one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4923_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [J2: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),aa_nat_nat(suc,J2))
     => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,A),aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B))),N)
      <=> $ite(N = zero_zero_nat,~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,numeral_numeral_int(bit0(one))),A),aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),B)),N)) ) ) ).

% bit_sum_mult_2_cases
tff(fact_4924_max_Oright__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.right_idem
tff(fact_4925_max_Oleft__idem,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ).

% max.left_idem
tff(fact_4926_max_Oidem,axiom,
    ! [A: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),A) = A ).

% max.idem
tff(fact_4927_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)
    <=> ( aa_rat_o(ord_less_eq_rat(B),A)
        & aa_rat_o(ord_less_eq_rat(C),A) ) ) ).

% max.bounded_iff
tff(fact_4928_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.bounded_iff
tff(fact_4929_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
        & aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.bounded_iff
tff(fact_4930_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb2
tff(fact_4931_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb2
tff(fact_4932_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb2
tff(fact_4933_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb1
tff(fact_4934_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb1
tff(fact_4935_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb1
tff(fact_4936_length__map,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn] : size_size_list_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)) = size_s6829681357464350627n_assn(Xs) ).

% length_map
tff(fact_4937_length__map,axiom,
    ! [F: fun_int_int,Xs: list_int] : size_size_list_int(map_int_int(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_4938_length__map,axiom,
    ! [F: fun_nat_int,Xs: list_nat] : size_size_list_int(map_nat_int(F,Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_4939_length__map,axiom,
    ! [F: fun_int_nat,Xs: list_int] : size_size_list_nat(map_int_nat(F,Xs)) = size_size_list_int(Xs) ).

% length_map
tff(fact_4940_length__map,axiom,
    ! [F: fun_nat_nat,Xs: list_nat] : size_size_list_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)) = size_size_list_nat(Xs) ).

% length_map
tff(fact_4941_max__less__iff__conj,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(ord_max_Code_integer(X,Y)),Z)
    <=> ( aa_Code_integer_o(ord_le6747313008572928689nteger(X),Z)
        & aa_Code_integer_o(ord_le6747313008572928689nteger(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4942_max__less__iff__conj,axiom,
    ! [X: rat,Y: rat,Z: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z)
    <=> ( aa_rat_o(ord_less_rat(X),Z)
        & aa_rat_o(ord_less_rat(Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4943_max__less__iff__conj,axiom,
    ! [X: 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,X),Y)),Z)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X),Z)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4944_max__less__iff__conj,axiom,
    ! [X: 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,X),Y)),Z)
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,X),Z)
        & aa_int_o(aa_int_fun_int_o(ord_less_int,Y),Z) ) ) ).

% max_less_iff_conj
tff(fact_4945_max_Oabsorb4,axiom,
    ! [A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(A),B)
     => ( ord_max_Code_integer(A,B) = B ) ) ).

% max.absorb4
tff(fact_4946_max_Oabsorb4,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(A),B)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4947_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,A),B)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb4
tff(fact_4948_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,A),B)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb4
tff(fact_4949_max_Oabsorb3,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
     => ( ord_max_Code_integer(A,B) = A ) ) ).

% max.absorb3
tff(fact_4950_max_Oabsorb3,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
     => ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4951_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb3
tff(fact_4952_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
     => ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb3
tff(fact_4953_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_4954_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_int,A3: list_int,B4: list_int] :
      ( ( size_size_list_int(A) = size_size_list_int(B) )
     => ( ( size_size_list_int(A3) = size_size_list_int(B4) )
       => ( ( zip_int_int(A,B) = zip_int_int(A3,B4) )
        <=> ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_4955_zip__eq__zip__same__len,axiom,
    ! [A: list_int,B: list_nat,A3: list_int,B4: list_nat] :
      ( ( size_size_list_int(A) = size_size_list_nat(B) )
     => ( ( size_size_list_int(A3) = size_size_list_nat(B4) )
       => ( ( zip_int_nat(A,B) = zip_int_nat(A3,B4) )
        <=> ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_4956_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_int,A3: list_nat,B4: list_int] :
      ( ( size_size_list_nat(A) = size_size_list_int(B) )
     => ( ( size_size_list_nat(A3) = size_size_list_int(B4) )
       => ( ( zip_nat_int(A,B) = zip_nat_int(A3,B4) )
        <=> ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_4957_zip__eq__zip__same__len,axiom,
    ! [A: list_nat,B: list_nat,A3: list_nat,B4: list_nat] :
      ( ( size_size_list_nat(A) = size_size_list_nat(B) )
     => ( ( size_size_list_nat(A3) = size_size_list_nat(B4) )
       => ( ( zip_nat_nat(A,B) = zip_nat_nat(A3,B4) )
        <=> ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_eq_zip_same_len
tff(fact_4958_length__enumerate,axiom,
    ! [N: nat,Xs: list_int] : size_s2970893825323803983at_int(enumerate_int(N,Xs)) = size_size_list_int(Xs) ).

% length_enumerate
tff(fact_4959_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] : size_s5460976970255530739at_nat(enumerate_nat(N,Xs)) = size_size_list_nat(Xs) ).

% length_enumerate
tff(fact_4960_length__rotate1,axiom,
    ! [Xs: list_int] : size_size_list_int(rotate1_int(Xs)) = size_size_list_int(Xs) ).

% length_rotate1
tff(fact_4961_length__rotate1,axiom,
    ! [Xs: list_nat] : size_size_list_nat(rotate1_nat(Xs)) = size_size_list_nat(Xs) ).

% length_rotate1
tff(fact_4962_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn(Xs) = zero_zero_nat )
    <=> ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
tff(fact_4963_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int(Xs) = zero_zero_nat )
    <=> ( Xs = nil_int ) ) ).

% length_0_conv
tff(fact_4964_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat(Xs) = zero_zero_nat )
    <=> ( Xs = nil_nat ) ) ).

% length_0_conv
tff(fact_4965_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_4966_max__0__1_I2_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),zero_zero_rat) = one_one_rat ).

% max_0_1(2)
tff(fact_4967_max__0__1_I2_J,axiom,
    ord_max_Code_integer(one_one_Code_integer,zero_z3403309356797280102nteger) = one_one_Code_integer ).

% max_0_1(2)
tff(fact_4968_max__0__1_I2_J,axiom,
    ord_max_Code_natural(one_one_Code_natural,zero_z2226904508553997617atural) = one_one_Code_natural ).

% max_0_1(2)
tff(fact_4969_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_4970_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_4971_max__0__1_I1_J,axiom,
    aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,zero_zero_rat),one_one_rat) = one_one_rat ).

% max_0_1(1)
tff(fact_4972_max__0__1_I1_J,axiom,
    ord_max_Code_integer(zero_z3403309356797280102nteger,one_one_Code_integer) = one_one_Code_integer ).

% max_0_1(1)
tff(fact_4973_max__0__1_I1_J,axiom,
    ord_max_Code_natural(zero_z2226904508553997617atural,one_one_Code_natural) = one_one_Code_natural ).

% max_0_1(1)
tff(fact_4974_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_4975_max__0__1_I6_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,numeral_numeral_nat(X)),one_one_nat) = numeral_numeral_nat(X) ).

% max_0_1(6)
tff(fact_4976_max__0__1_I6_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,numeral_numeral_int(X)),one_one_int) = numeral_numeral_int(X) ).

% max_0_1(6)
tff(fact_4977_max__0__1_I6_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,numeral_numeral_rat(X)),one_one_rat) = numeral_numeral_rat(X) ).

% max_0_1(6)
tff(fact_4978_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_Code_integer(numera6620942414471956472nteger(X),one_one_Code_integer) = numera6620942414471956472nteger(X) ).

% max_0_1(6)
tff(fact_4979_max__0__1_I6_J,axiom,
    ! [X: num] : ord_max_Code_natural(numera5444537566228673987atural(X),one_one_Code_natural) = numera5444537566228673987atural(X) ).

% max_0_1(6)
tff(fact_4980_max__0__1_I5_J,axiom,
    ! [X: num] : aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,one_one_nat),numeral_numeral_nat(X)) = numeral_numeral_nat(X) ).

% max_0_1(5)
tff(fact_4981_max__0__1_I5_J,axiom,
    ! [X: num] : aa_int_int(aa_int_fun_int_int(ord_max_int,one_one_int),numeral_numeral_int(X)) = numeral_numeral_int(X) ).

% max_0_1(5)
tff(fact_4982_max__0__1_I5_J,axiom,
    ! [X: num] : aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,one_one_rat),numeral_numeral_rat(X)) = numeral_numeral_rat(X) ).

% max_0_1(5)
tff(fact_4983_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_Code_integer(one_one_Code_integer,numera6620942414471956472nteger(X)) = numera6620942414471956472nteger(X) ).

% max_0_1(5)
tff(fact_4984_max__0__1_I5_J,axiom,
    ! [X: num] : ord_max_Code_natural(one_one_Code_natural,numera5444537566228673987atural(X)) = numera5444537566228673987atural(X) ).

% max_0_1(5)
tff(fact_4985_length__splice,axiom,
    ! [Xs: list_int,Ys: list_int] : size_size_list_int(splice_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_splice
tff(fact_4986_length__splice,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_size_list_nat(splice_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_splice
tff(fact_4987_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] : size_s5157815400016825771nt_int(product_int_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4988_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] : size_s7647898544948552527nt_nat(product_int_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_int(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_4989_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] : size_s2970893825323803983at_int(product_nat_int(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_int(Ys)) ).

% length_product
tff(fact_4990_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] : size_s5460976970255530739at_nat(product_nat_nat(Xs,Ys)) = aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,size_size_list_nat(Xs)),size_size_list_nat(Ys)) ).

% length_product
tff(fact_4991_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_s6829681357464350627n_assn(Xs))
    <=> ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
tff(fact_4992_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_int(Xs))
    <=> ( Xs != nil_int ) ) ).

% length_greater_0_conv
tff(fact_4993_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,zero_zero_nat),size_size_list_nat(Xs))
    <=> ( Xs != nil_nat ) ) ).

% length_greater_0_conv
tff(fact_4994_nth__map,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,F: fun_Pr3816910165196019820n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_s6829681357464350627n_assn(Xs))
     => ( aa_nat_assn(nth_assn(aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs)),N) = aa_Pro2979818503705714077n_assn(F,aa_nat3152888703114873891n_assn(nth_Pr1769885009046257848n_assn(Xs),N)) ) ) ).

% nth_map
tff(fact_4995_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: fun_int_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( aa_nat_nat(nth_nat(map_int_nat(F,Xs)),N) = aa_int_nat(F,aa_nat_int(nth_int(Xs),N)) ) ) ).

% nth_map
tff(fact_4996_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: fun_int_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_int(Xs))
     => ( aa_nat_int(nth_int(map_int_int(F,Xs)),N) = aa_int_int(F,aa_nat_int(nth_int(Xs),N)) ) ) ).

% nth_map
tff(fact_4997_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: fun_nat_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( aa_nat_int(nth_int(map_nat_int(F,Xs)),N) = aa_nat_int(F,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% nth_map
tff(fact_4998_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: fun_nat_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),size_size_list_nat(Xs))
     => ( aa_nat_nat(nth_nat(aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs)),N) = aa_nat_nat(F,aa_nat_nat(nth_nat(Xs),N)) ) ) ).

% nth_map
tff(fact_4999_drop__bit__of__1,axiom,
    ! [N: nat] : bit_se3928097537394005634nteger(N,one_one_Code_integer) = zero_n356916108424825756nteger(N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_5000_drop__bit__of__1,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),one_one_nat) = zero_n2687167440665602831ol_nat(N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_5001_drop__bit__of__1,axiom,
    ! [N: nat] : aa_int_int(bit_se8568078237143864401it_int(N),one_one_int) = zero_n2684676970156552555ol_int(N = zero_zero_nat) ).

% drop_bit_of_1
tff(fact_5002_map__fst__zip,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( map_Pr4307537318738222789at_int(product_fst_int_nat,zip_int_nat(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5003_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( map_Pr6125295803703450217nt_nat(product_fst_nat_int,zip_nat_int(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5004_map__fst__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( map_Pr3938374229010428429at_nat(product_fst_nat_nat,zip_nat_nat(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5005_map__fst__zip,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( size_s7011499401410611007t_unit(Xs) = size_s3254054031482475050et_nat(Ys) )
     => ( map_Pr7042224834074979589t_unit(produc1824681642469235216et_nat,zip_he6711929980790699996et_nat(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5006_map__fst__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( map_Pr6494458893431244577nt_int(product_fst_int_int,zip_int_int(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5007_map__fst__zip,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( size_size_list_assn(Xs) = size_size_list_assn(Ys) )
     => ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),zip_assn_assn(Xs,Ys)) = Xs ) ) ).

% map_fst_zip
tff(fact_5008_map__snd__zip,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( map_Pr4310027789247273065at_nat(product_snd_int_nat,zip_int_nat(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5009_map__snd__zip,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( map_Pr6122805333194399941nt_int(product_snd_nat_int,zip_nat_int(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5010_map__snd__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat(Xs) = size_size_list_nat(Ys) )
     => ( map_Pr3938374229010428429at_nat(product_snd_nat_nat,zip_nat_nat(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5011_map__snd__zip,axiom,
    ! [Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( ( size_s7011499401410611007t_unit(Xs) = size_s3254054031482475050et_nat(Ys) )
     => ( map_Pr4871828589360411470et_nat(produc8586169260539613262et_nat,zip_he6711929980790699996et_nat(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5012_map__snd__zip,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( map_Pr6494458893431244577nt_int(product_snd_int_int,zip_int_int(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5013_map__snd__zip,axiom,
    ! [Xs: list_assn,Ys: list_assn] :
      ( ( size_size_list_assn(Xs) = size_size_list_assn(Ys) )
     => ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),zip_assn_assn(Xs,Ys)) = Ys ) ) ).

% map_snd_zip
tff(fact_5014_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_int(Xs)),one_one_nat)
     => ( rotate1_int(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5015_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,size_size_list_nat(Xs)),one_one_nat)
     => ( rotate1_nat(Xs) = Xs ) ) ).

% rotate1_length01
tff(fact_5016_length__ge__1__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_s6829681357464350627n_assn(L))
    <=> ( L != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
tff(fact_5017_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_int(L))
    <=> ( L != nil_int ) ) ).

% length_ge_1_conv
tff(fact_5018_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,zero_zero_nat)),size_size_list_nat(L))
    <=> ( L != nil_nat ) ) ).

% length_ge_1_conv
tff(fact_5019_nth__zip,axiom,
    ! [I2: nat,Xs: list_h2745205591237434579t_unit,Ys: list_set_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7011499401410611007t_unit(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s3254054031482475050et_nat(Ys))
       => ( nth_Pr7683458668995092411et_nat(zip_he6711929980790699996et_nat(Xs,Ys),I2) = produc7507926704131184380et_nat(nth_he7452997726511547988t_unit(Xs,I2),aa_nat_set_nat(nth_set_nat(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_5020_nth__zip,axiom,
    ! [I2: nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P9062070895058802706et_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7422308124397347553_nat_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s2864271983995610790et_nat(Ys))
       => ( nth_Pr3203592224402014297et_nat(zip_fu1498620252286093946et_nat(Xs,Ys),I2) = produc6017135503889078682et_nat(nth_fu1272478342191624950_nat_o(Xs,I2),nth_Pr7683458668995092411et_nat(Ys,I2)) ) ) ) ).

% nth_zip
tff(fact_5021_nth__zip,axiom,
    ! [I2: nat,Xs: list_f5222602229612168821_nat_o,Ys: list_P8836981409019564976et_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s7422308124397347553_nat_o(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_s6978193692292872260et_nat(Ys))
       => ( nth_Pr5532225396715843703et_nat(zip_fu1756369540273021848et_nat(Xs,Ys),I2) = produc7291855603632349880et_nat(nth_fu1272478342191624950_nat_o(Xs,I2),nth_Pr3203592224402014297et_nat(Ys,I2)) ) ) ) ).

% nth_zip
tff(fact_5022_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys: list_int] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Ys))
       => ( nth_Pr4439495888332055232nt_int(zip_int_int(Xs,Ys),I2) = product_Pair_int_int(aa_nat_int(nth_int(Xs),I2),aa_nat_int(nth_int(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_5023_nth__zip,axiom,
    ! [I2: nat,Xs: list_int,Ys: list_nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_int(Xs))
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Ys))
       => ( nth_Pr8617346907841251940nt_nat(zip_int_nat(Xs,Ys),I2) = product_Pair_int_nat(aa_nat_int(nth_int(Xs),I2),aa_nat_nat(nth_nat(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_5024_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_int] :
      ( 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,I2),size_size_list_int(Ys))
       => ( nth_Pr3440142176431000676at_int(zip_nat_int(Xs,Ys),I2) = product_Pair_nat_int(aa_nat_nat(nth_nat(Xs),I2),aa_nat_int(nth_int(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_5025_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_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,I2),size_size_list_nat(Ys))
       => ( nth_Pr7617993195940197384at_nat(zip_nat_nat(Xs,Ys),I2) = product_Pair_nat_nat(aa_nat_nat(nth_nat(Xs),I2),aa_nat_nat(nth_nat(Ys),I2)) ) ) ) ).

% nth_zip
tff(fact_5026_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_5027_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_5028_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] : size_size_list_int(Xs2) = N ).

% Ex_list_of_length
tff(fact_5029_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] : size_size_list_nat(Xs2) = N ).

% Ex_list_of_length
tff(fact_5030_max_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),C)) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.left_commute
tff(fact_5031_max_Ocommute,axiom,
    ! [A: int,B: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = aa_int_int(aa_int_fun_int_int(ord_max_int,B),A) ).

% max.commute
tff(fact_5032_max_Oassoc,axiom,
    ! [A: int,B: int,C: int] : aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)),C) = aa_int_int(aa_int_fun_int_int(ord_max_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)) ).

% max.assoc
tff(fact_5033_pair__list__split,axiom,
    ! [L: list_P5707943133018811711nt_int] :
      ~ ! [L1: list_int,L2: list_int] :
          ( ( L = zip_int_int(L1,L2) )
         => ( ( size_size_list_int(L1) = size_size_list_int(L2) )
           => ( size_s5157815400016825771nt_int(L) != size_size_list_int(L2) ) ) ) ).

% pair_list_split
tff(fact_5034_pair__list__split,axiom,
    ! [L: list_P8198026277950538467nt_nat] :
      ~ ! [L1: list_int,L2: list_nat] :
          ( ( L = zip_int_nat(L1,L2) )
         => ( ( size_size_list_int(L1) = size_size_list_nat(L2) )
           => ( size_s7647898544948552527nt_nat(L) != size_size_list_nat(L2) ) ) ) ).

% pair_list_split
tff(fact_5035_pair__list__split,axiom,
    ! [L: list_P3521021558325789923at_int] :
      ~ ! [L1: list_nat,L2: list_int] :
          ( ( L = zip_nat_int(L1,L2) )
         => ( ( size_size_list_nat(L1) = size_size_list_int(L2) )
           => ( size_s2970893825323803983at_int(L) != size_size_list_int(L2) ) ) ) ).

% pair_list_split
tff(fact_5036_pair__list__split,axiom,
    ! [L: list_P6011104703257516679at_nat] :
      ~ ! [L1: list_nat,L2: list_nat] :
          ( ( L = zip_nat_nat(L1,L2) )
         => ( ( size_size_list_nat(L1) = size_size_list_nat(L2) )
           => ( size_s5460976970255530739at_nat(L) != size_size_list_nat(L2) ) ) ) ).

% pair_list_split
tff(fact_5037_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: code_integer,N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(A),N)
    <=> ( aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(bit_se3949692690581998587nteger,bit_se3928097537394005634nteger(N,A)),one_one_Code_integer) = one_one_Code_integer ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5038_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,A),N)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(bit_se727722235901077358nd_nat,aa_nat_nat(aa_nat_fun_nat_nat(bit_se8570568707652914677it_nat,N),A)),one_one_nat) = one_one_nat ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5039_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: int,N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,A),N)
    <=> ( aa_int_int(aa_int_fun_int_int(bit_se725231765392027082nd_int,aa_int_int(bit_se8568078237143864401it_int(N),A)),one_one_int) = one_one_int ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_5040_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),B)
     => aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI2
tff(fact_5041_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI2
tff(fact_5042_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI2
tff(fact_5043_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),A)
     => aa_rat_o(ord_less_eq_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.coboundedI1
tff(fact_5044_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.coboundedI1
tff(fact_5045_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.coboundedI1
tff(fact_5046_max_Oabsorb__iff2,axiom,
    ! [A: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(A),B)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5047_max_Oabsorb__iff2,axiom,
    ! [A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),B)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5048_max_Oabsorb__iff2,axiom,
    ! [A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = B ) ) ).

% max.absorb_iff2
tff(fact_5049_max_Oabsorb__iff1,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
    <=> ( aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5050_max_Oabsorb__iff1,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5051_max_Oabsorb__iff1,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) = A ) ) ).

% max.absorb_iff1
tff(fact_5052_le__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(ord_less_eq_rat(Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y))
    <=> ( aa_rat_o(ord_less_eq_rat(Z),X)
        | aa_rat_o(ord_less_eq_rat(Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5053_le__max__iff__disj,axiom,
    ! [Z: nat,X: 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,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5054_le__max__iff__disj,axiom,
    ! [Z: int,X: 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,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_eq_int,Z),Y) ) ) ).

% le_max_iff_disj
tff(fact_5055_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : aa_rat_o(ord_less_eq_rat(B),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ).

% max.cobounded2
tff(fact_5056_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded2
tff(fact_5057_max_Ocobounded2,axiom,
    ! [B: int,A: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded2
tff(fact_5058_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : aa_rat_o(ord_less_eq_rat(A),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ).

% max.cobounded1
tff(fact_5059_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,A),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ).

% max.cobounded1
tff(fact_5060_max_Ocobounded1,axiom,
    ! [A: int,B: int] : aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ).

% max.cobounded1
tff(fact_5061_max_Oorder__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
    <=> ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.order_iff
tff(fact_5062_max_Oorder__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
    <=> ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.order_iff
tff(fact_5063_max_Oorder__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
    <=> ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.order_iff
tff(fact_5064_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( aa_rat_o(ord_less_eq_rat(C),A)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_5065_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A) ) ) ).

% max.boundedI
tff(fact_5066_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A) ) ) ).

% max.boundedI
tff(fact_5067_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)
     => ~ ( aa_rat_o(ord_less_eq_rat(B),A)
         => ~ aa_rat_o(ord_less_eq_rat(C),A) ) ) ).

% max.boundedE
tff(fact_5068_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A) ) ) ).

% max.boundedE
tff(fact_5069_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A) ) ) ).

% max.boundedE
tff(fact_5070_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
     => aa_rat_o(ord_less_eq_rat(B),A) ) ).

% max.orderI
tff(fact_5071_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A) ) ).

% max.orderI
tff(fact_5072_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
     => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ).

% max.orderI
tff(fact_5073_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_eq_rat(B),A)
     => ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) ) ) ).

% max.orderE
tff(fact_5074_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,B),A)
     => ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) ) ) ).

% max.orderE
tff(fact_5075_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A)
     => ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) ) ) ).

% max.orderE
tff(fact_5076_max_Omono,axiom,
    ! [C: rat,A: rat,D2: rat,B: rat] :
      ( aa_rat_o(ord_less_eq_rat(C),A)
     => ( aa_rat_o(ord_less_eq_rat(D2),B)
       => aa_rat_o(ord_less_eq_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,C),D2)),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ) ).

% max.mono
tff(fact_5077_max_Omono,axiom,
    ! [C: nat,A: nat,D2: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,C),A)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,D2),B)
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,C),D2)),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ) ).

% max.mono
tff(fact_5078_max_Omono,axiom,
    ! [C: int,A: int,D2: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,C),A)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,D2),B)
       => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(ord_max_int,C),D2)),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ) ).

% max.mono
tff(fact_5079_max_Ostrict__coboundedI2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),B)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI2
tff(fact_5080_max_Ostrict__coboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(C),B)
     => aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5081_max_Ostrict__coboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),B)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5082_max_Ostrict__coboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),B)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI2
tff(fact_5083_max_Ostrict__coboundedI1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(C),A)
     => aa_Code_integer_o(ord_le6747313008572928689nteger(C),ord_max_Code_integer(A,B)) ) ).

% max.strict_coboundedI1
tff(fact_5084_max_Ostrict__coboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( aa_rat_o(ord_less_rat(C),A)
     => aa_rat_o(ord_less_rat(C),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5085_max_Ostrict__coboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A)
     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5086_max_Ostrict__coboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,C),A)
     => aa_int_o(aa_int_fun_int_o(ord_less_int,C),aa_int_int(aa_int_fun_int_int(ord_max_int,A),B)) ) ).

% max.strict_coboundedI1
tff(fact_5087_max_Ostrict__order__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
    <=> ( ( A = ord_max_Code_integer(A,B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5088_max_Ostrict__order__iff,axiom,
    ! [B: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(B),A)
    <=> ( ( A = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5089_max_Ostrict__order__iff,axiom,
    ! [B: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
    <=> ( ( A = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5090_max_Ostrict__order__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
    <=> ( ( A = aa_int_int(aa_int_fun_int_int(ord_max_int,A),B) )
        & ( A != B ) ) ) ).

% max.strict_order_iff
tff(fact_5091_max_Ostrict__boundedE,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(ord_max_Code_integer(B,C)),A)
     => ~ ( aa_Code_integer_o(ord_le6747313008572928689nteger(B),A)
         => ~ aa_Code_integer_o(ord_le6747313008572928689nteger(C),A) ) ) ).

% max.strict_boundedE
tff(fact_5092_max_Ostrict__boundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( aa_rat_o(ord_less_rat(aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,B),C)),A)
     => ~ ( aa_rat_o(ord_less_rat(B),A)
         => ~ aa_rat_o(ord_less_rat(C),A) ) ) ).

% max.strict_boundedE
tff(fact_5093_max_Ostrict__boundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,B),C)),A)
     => ~ ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,B),A)
         => ~ aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5094_max_Ostrict__boundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(ord_max_int,B),C)),A)
     => ~ ( aa_int_o(aa_int_fun_int_o(ord_less_int,B),A)
         => ~ aa_int_o(aa_int_fun_int_o(ord_less_int,C),A) ) ) ).

% max.strict_boundedE
tff(fact_5095_less__max__iff__disj,axiom,
    ! [Z: code_integer,X: code_integer,Y: code_integer] :
      ( aa_Code_integer_o(ord_le6747313008572928689nteger(Z),ord_max_Code_integer(X,Y))
    <=> ( aa_Code_integer_o(ord_le6747313008572928689nteger(Z),X)
        | aa_Code_integer_o(ord_le6747313008572928689nteger(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5096_less__max__iff__disj,axiom,
    ! [Z: rat,X: rat,Y: rat] :
      ( aa_rat_o(ord_less_rat(Z),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y))
    <=> ( aa_rat_o(ord_less_rat(Z),X)
        | aa_rat_o(ord_less_rat(Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5097_less__max__iff__disj,axiom,
    ! [Z: nat,X: 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,X),Y))
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),X)
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5098_less__max__iff__disj,axiom,
    ! [Z: int,X: 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,X),Y))
    <=> ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z),X)
        | aa_int_o(aa_int_fun_int_o(ord_less_int,Z),Y) ) ) ).

% less_max_iff_disj
tff(fact_5099_max__add__distrib__right,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),ord_max_Code_integer(Y,Z)) = ord_max_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Y),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z)) ).

% max_add_distrib_right
tff(fact_5100_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,Y),Z)) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Y)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)) ).

% max_add_distrib_right
tff(fact_5101_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,Y),Z)) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Y)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)) ).

% max_add_distrib_right
tff(fact_5102_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,X),aa_int_int(aa_int_fun_int_int(ord_max_int,Y),Z)) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Y)),aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)) ).

% max_add_distrib_right
tff(fact_5103_max__add__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,ord_max_Code_integer(X,Y)),Z) = ord_max_Code_integer(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,X),Z),aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,Y),Z)) ).

% max_add_distrib_left
tff(fact_5104_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z: rat] : aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,X),Y)),Z) = aa_rat_rat(aa_rat_fun_rat_rat(ord_max_rat,aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,X),Z)),aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,Y),Z)) ).

% max_add_distrib_left
tff(fact_5105_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] : aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,X),Y)),Z) = aa_nat_nat(aa_nat_fun_nat_nat(ord_max_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,X),Z)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Y),Z)) ).

% max_add_distrib_left
tff(fact_5106_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X),Z)),aa_int_int(aa_int_fun_int_int(plus_plus_int,Y),Z)) ).

% max_add_distrib_left
tff(fact_5107_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] : aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(aa_int_fun_int_int(ord_max_int,X),Y)),Z) = aa_int_int(aa_int_fun_int_int(ord_max_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X),Z)),aa_int_int(aa_int_fun_int_int(minus_minus_int,Y),Z)) ).

% max_diff_distrib_left
tff(fact_5108_max__diff__distrib__left,axiom,
    ! [X: code_integer,Y: code_integer,Z: code_integer] : minus_8373710615458151222nteger(ord_max_Code_integer(X,Y),Z) = ord_max_Code_integer(minus_8373710615458151222nteger(X,Z),minus_8373710615458151222nteger(Y,Z)) ).

% max_diff_distrib_left
tff(fact_5109_sup__max,axiom,
    sup_sup_int = ord_max_int ).

% sup_max
tff(fact_5110_length__induct,axiom,
    ! [Pa: fun_list_int_o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys5: list_int] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(Ys5)),size_size_list_int(Xs2))
             => aa_list_int_o(Pa,Ys5) )
         => aa_list_int_o(Pa,Xs2) )
     => aa_list_int_o(Pa,Xs) ) ).

% length_induct
tff(fact_5111_length__induct,axiom,
    ! [Pa: fun_list_nat_o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys5: list_nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(Ys5)),size_size_list_nat(Xs2))
             => aa_list_nat_o(Pa,Ys5) )
         => aa_list_nat_o(Pa,Xs2) )
     => aa_list_nat_o(Pa,Xs) ) ).

% length_induct
tff(fact_5112_finite__maxlen,axiom,
    ! [M: set_list_int] :
      ( finite3922522038869484883st_int(M)
     => ? [N3: nat] :
        ! [X3: list_int] :
          ( member_list_int(X3,M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_int(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_5113_finite__maxlen,axiom,
    ! [M: set_list_nat] :
      ( finite8100373058378681591st_nat(M)
     => ? [N3: nat] :
        ! [X3: list_nat] :
          ( member_list_nat(X3,M)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,size_size_list_nat(X3)),N3) ) ) ).

% finite_maxlen
tff(fact_5114_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G),Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5115_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G: fun_int_assn,Ys: list_int] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = map_int_assn(G,Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5116_map__eq__imp__length__eq,axiom,
    ! [F: fun_Pr3816910165196019820n_assn,Xs: list_P8527749157015355191n_assn,G: fun_nat_assn,Ys: list_nat] :
      ( ( aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,F),Xs) = map_nat_assn(G,Ys) )
     => ( size_s6829681357464350627n_assn(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5117_map__eq__imp__length__eq,axiom,
    ! [F: fun_int_assn,Xs: list_int,G: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_int_assn(F,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G),Ys) )
     => ( size_size_list_int(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5118_map__eq__imp__length__eq,axiom,
    ! [F: fun_int_nat,Xs: list_int,G: fun_nat_nat,Ys: list_nat] :
      ( ( map_int_nat(F,Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,G),Ys) )
     => ( size_size_list_int(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5119_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_assn,Xs: list_nat,G: fun_Pr3816910165196019820n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_nat_assn(F,Xs) = aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,G),Ys) )
     => ( size_size_list_nat(Xs) = size_s6829681357464350627n_assn(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5120_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_int_nat,Ys: list_int] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = map_int_nat(G,Ys) )
     => ( size_size_list_nat(Xs) = size_size_list_int(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5121_map__eq__imp__length__eq,axiom,
    ! [F: fun_nat_nat,Xs: list_nat,G: fun_nat_nat,Ys: list_nat] :
      ( ( aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,F),Xs) = aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,G),Ys) )
     => ( size_size_list_nat(Xs) = size_size_list_nat(Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_5122_max_Oabel__semigroup__axioms,axiom,
    abel_semigroup_int(ord_max_int) ).

% max.abel_semigroup_axioms
tff(fact_5123_zip__inj,axiom,
    ! [A: list_int,B: list_int,A3: list_int,B4: list_int] :
      ( ( size_size_list_int(A) = size_size_list_int(B) )
     => ( ( size_size_list_int(A3) = size_size_list_int(B4) )
       => ( ( zip_int_int(A,B) = zip_int_int(A3,B4) )
         => ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_inj
tff(fact_5124_zip__inj,axiom,
    ! [A: list_int,B: list_nat,A3: list_int,B4: list_nat] :
      ( ( size_size_list_int(A) = size_size_list_nat(B) )
     => ( ( size_size_list_int(A3) = size_size_list_nat(B4) )
       => ( ( zip_int_nat(A,B) = zip_int_nat(A3,B4) )
         => ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_inj
tff(fact_5125_zip__inj,axiom,
    ! [A: list_nat,B: list_int,A3: list_nat,B4: list_int] :
      ( ( size_size_list_nat(A) = size_size_list_int(B) )
     => ( ( size_size_list_nat(A3) = size_size_list_int(B4) )
       => ( ( zip_nat_int(A,B) = zip_nat_int(A3,B4) )
         => ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_inj
tff(fact_5126_zip__inj,axiom,
    ! [A: list_nat,B: list_nat,A3: list_nat,B4: list_nat] :
      ( ( size_size_list_nat(A) = size_size_list_nat(B) )
     => ( ( size_size_list_nat(A3) = size_size_list_nat(B4) )
       => ( ( zip_nat_nat(A,B) = zip_nat_nat(A3,B4) )
         => ( ( A = A3 )
            & ( B = B4 ) ) ) ) ) ).

% zip_inj
tff(fact_5127_max_Osemilattice__axioms,axiom,
    semilattice_int(ord_max_int) ).

% max.semilattice_axioms
tff(fact_5128_max_Osemigroup__axioms,axiom,
    semigroup_int(ord_max_int) ).

% max.semigroup_axioms
tff(fact_5129_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5130_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5131_bit__1__iff,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),N)
    <=> ( N = zero_zero_nat ) ) ).

% bit_1_iff
tff(fact_5132_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_5133_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_5134_not__bit__1__Suc,axiom,
    ! [N: nat] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),aa_nat_nat(suc,N)) ).

% not_bit_1_Suc
tff(fact_5135_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(bit_se9216721137139052372nteger(one_one_Code_integer),numeral_numeral_nat(N)) ).

% bit_numeral_simps(1)
tff(fact_5136_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(aa_nat_fun_nat_o(bit_se1148574629649215175it_nat,one_one_nat),numeral_numeral_nat(N)) ).

% bit_numeral_simps(1)
tff(fact_5137_bit__numeral__simps_I1_J,axiom,
    ! [N: num] : ~ aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,one_one_int),numeral_numeral_nat(N)) ).

% bit_numeral_simps(1)
tff(fact_5138_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_int,Pa: fun_li2369922317605390092_int_o] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis7635411640666648821_int_o(Pa,nil_int),nil_int),nil_int),nil_int)
           => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( size_size_list_int(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis7635411640666648821_int_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis7635411640666648821_int_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs3)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis7635411640666648821_int_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5139_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_int,Ws: list_nat,Pa: fun_li1129935163125380200_nat_o] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis8500848182373993553_nat_o(Pa,nil_int),nil_int),nil_int),nil_nat)
           => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_int(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis8500848182373993553_nat_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis8500848182373993553_nat_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs3)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis8500848182373993553_nat_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5140_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_int,Pa: fun_li7530032041894550632_int_o] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis1028340864530234420_int_o(aa_lis3920731331146833_int_o(Pa,nil_int),nil_int),nil_nat),nil_int)
           => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int,Z2: nat,Zs3: list_nat,W3: int,Ws2: list_int] :
                  ( ( size_size_list_int(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_nat(Zs3) )
                   => ( ( size_size_list_nat(Zs3) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis1028340864530234420_int_o(aa_lis3920731331146833_int_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis1028340864530234420_int_o(aa_lis3920731331146833_int_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs3)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis1028340864530234420_int_o(aa_lis3920731331146833_int_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5141_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_int,Zs: list_nat,Ws: list_nat,Pa: fun_li6290044887414540740_nat_o] :
      ( ( size_size_list_int(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis956390372307273104_nat_o(aa_lis869357273038491565_nat_o(Pa,nil_int),nil_int),nil_nat),nil_nat)
           => ( ! [X2: int,Xs2: list_int,Y2: int,Ys3: list_int,Z2: nat,Zs3: list_nat,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_int(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_nat(Zs3) )
                   => ( ( size_size_list_nat(Zs3) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis956390372307273104_nat_o(aa_lis869357273038491565_nat_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis956390372307273104_nat_o(aa_lis869357273038491565_nat_o(Pa,cons_int(X2,Xs2)),cons_int(Y2,Ys3)),cons_nat(Z2,Zs3)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis956390372307273104_nat_o(aa_lis869357273038491565_nat_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5142_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_int,Pa: fun_li5566396085233943656_int_o] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis2666369435012445748_int_o(aa_lis9035863069498285137_int_o(Pa,nil_int),nil_nat),nil_int),nil_int)
           => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( size_size_list_int(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis2666369435012445748_int_o(aa_lis9035863069498285137_int_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis2666369435012445748_int_o(aa_lis9035863069498285137_int_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)),cons_int(Z2,Zs3)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis2666369435012445748_int_o(aa_lis9035863069498285137_int_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5143_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_int,Ws: list_nat,Pa: fun_li4326408930753933764_nat_o] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis2594418942789484432_nat_o(aa_lis677927574350854061_nat_o(Pa,nil_int),nil_nat),nil_int),nil_nat)
           => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_int(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis2594418942789484432_nat_o(aa_lis677927574350854061_nat_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis2594418942789484432_nat_o(aa_lis677927574350854061_nat_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)),cons_int(Z2,Zs3)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis2594418942789484432_nat_o(aa_lis677927574350854061_nat_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5144_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_int,Pa: fun_li1503133772668328388_int_o] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis2428792293361870736_int_o(aa_lis1404372160162783149_int_o(Pa,nil_int),nil_nat),nil_nat),nil_int)
           => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat,Z2: nat,Zs3: list_nat,W3: int,Ws2: list_int] :
                  ( ( size_size_list_int(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_nat(Zs3) )
                   => ( ( size_size_list_nat(Zs3) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis2428792293361870736_int_o(aa_lis1404372160162783149_int_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis2428792293361870736_int_o(aa_lis1404372160162783149_int_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)),cons_nat(Z2,Zs3)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis7535912258512046743_int_o(aa_lis2428792293361870736_int_o(aa_lis1404372160162783149_int_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5145_list__induct4,axiom,
    ! [Xs: list_int,Ys: list_nat,Zs: list_nat,Ws: list_nat,Pa: fun_li263146618188318496_nat_o] :
      ( ( size_size_list_int(Xs) = size_size_list_nat(Ys) )
     => ( ( size_size_list_nat(Ys) = size_size_list_nat(Zs) )
       => ( ( size_size_list_nat(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis2356841801138909420_nat_o(aa_lis2269808701870127881_nat_o(Pa,nil_int),nil_nat),nil_nat),nil_nat)
           => ( ! [X2: int,Xs2: list_int,Y2: nat,Ys3: list_nat,Z2: nat,Zs3: list_nat,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_int(Xs2) = size_size_list_nat(Ys3) )
                 => ( ( size_size_list_nat(Ys3) = size_size_list_nat(Zs3) )
                   => ( ( size_size_list_nat(Zs3) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis2356841801138909420_nat_o(aa_lis2269808701870127881_nat_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis2356841801138909420_nat_o(aa_lis2269808701870127881_nat_o(Pa,cons_int(X2,Xs2)),cons_nat(Y2,Ys3)),cons_nat(Z2,Zs3)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis557685189327716851_nat_o(aa_lis2356841801138909420_nat_o(aa_lis2269808701870127881_nat_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5146_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_int,Pa: fun_li6224140955532782696_int_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_int(Ws) )
         => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis2266258241739265617_int_o(Pa,nil_nat),nil_int),nil_int),nil_int)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: int,Ws2: list_int] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_int(Ws2) )
                     => ( aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis2266258241739265617_int_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis2266258241739265617_int_o(Pa,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs3)),cons_int(W3,Ws2)) ) ) ) )
             => aa_list_int_o(aa_lis7773489400162621755_int_o(aa_lis1265918006180809432_int_o(aa_lis2266258241739265617_int_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5147_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs: list_int,Ws: list_nat,Pa: fun_li4984153801052772804_nat_o] :
      ( ( size_size_list_nat(Xs) = size_size_list_int(Ys) )
     => ( ( size_size_list_int(Ys) = size_size_list_int(Zs) )
       => ( ( size_size_list_int(Zs) = size_size_list_nat(Ws) )
         => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis3131694783446610349_nat_o(Pa,nil_nat),nil_int),nil_int),nil_nat)
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys3: list_int,Z2: int,Zs3: list_int,W3: nat,Ws2: list_nat] :
                  ( ( size_size_list_nat(Xs2) = size_size_list_int(Ys3) )
                 => ( ( size_size_list_int(Ys3) = size_size_list_int(Zs3) )
                   => ( ( size_size_list_int(Zs3) = size_size_list_nat(Ws2) )
                     => ( aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis3131694783446610349_nat_o(Pa,Xs2),Ys3),Zs3),Ws2)
                       => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis3131694783446610349_nat_o(Pa,cons_nat(X2,Xs2)),cons_int(Y2,Ys3)),cons_int(Z2,Zs3)),cons_nat(W3,Ws2)) ) ) ) )
             => aa_list_nat_o(aa_lis795262330978291863_nat_o(aa_lis1193967513957848116_nat_o(aa_lis3131694783446610349_nat_o(Pa,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5148_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_5149_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_5150_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N3),M3)
             => ( aa_nat_o(aa_int_fun_nat_o(bit_se1146084159140164899it_int,K),M3)
              <=> 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_5151_nat__numeral__diff__1,axiom,
    ! [V2: num] : aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,numeral_numeral_nat(V2)),one_one_nat) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,numeral_numeral_int(V2)),one_one_int)) ).

% nat_numeral_diff_1
tff(fact_5152_nat__1,axiom,
    aa_int_nat(nat2,one_one_int) = aa_nat_nat(suc,zero_zero_nat) ).

% nat_1
tff(fact_5153_length__upto,axiom,
    ! [I2: int,J: int] : size_size_list_int(aa_int_list_int(upto(I2),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),I2)),one_one_int)) ).

% length_upto
tff(fact_5154_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_5155_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( aa_list_nat_o(sorted_wrt_nat(ord_less_nat),Ns)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),size_size_list_nat(Ns))
       => aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),aa_nat_nat(nth_nat(Ns),I2)) ) ) ).

% sorted_wrt_less_idx
tff(fact_5156_sorted__upto,axiom,
    ! [M2: int,N: int] : aa_list_int_o(sorted_wrt_int(ord_less_eq_int),aa_int_list_int(upto(M2),N)) ).

% sorted_upto
tff(fact_5157_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : aa_list_int_o(sorted_wrt_int(ord_less_int),aa_int_list_int(upto(I2),J)) ).

% sorted_wrt_upto
tff(fact_5158_nat__one__as__int,axiom,
    one_one_nat = aa_int_nat(nat2,one_one_int) ).

% nat_one_as_int
tff(fact_5159_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_5160_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_list_int(upto(I2),J) = append_int(aa_int_list_int(upto(I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,nil_int)) ) ) ).

% upto_rec2
tff(fact_5161_normalize__denom__zero,axiom,
    ! [P4: int] : normalize(product_Pair_int_int(P4,zero_zero_int)) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% normalize_denom_zero
tff(fact_5162_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2)
     => ( aa_nat_list_nat(upt(I2),J) = nil_nat ) ) ).

% upt_conv_Nil
tff(fact_5163_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( aa_nat_list_nat(upt(I2),J) = nil_nat )
    <=> ( ( J = zero_zero_nat )
        | aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,J),I2) ) ) ).

% upt_eq_Nil_conv
tff(fact_5164_length__upt,axiom,
    ! [I2: nat,J: nat] : size_size_list_nat(aa_nat_list_nat(upt(I2),J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2) ).

% length_upt
tff(fact_5165_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      aa_nat_list_nat(upt(numeral_numeral_nat(M2)),numeral_numeral_nat(N)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,numeral_numeral_nat(M2)),numeral_numeral_nat(N)),cons_nat(numeral_numeral_nat(M2),aa_nat_list_nat(upt(aa_nat_nat(suc,numeral_numeral_nat(M2))),numeral_numeral_nat(N))),nil_nat) ).

% upt_rec_numeral
tff(fact_5166_nth__upt,axiom,
    ! [I2: 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,I2),K)),J)
     => ( aa_nat_nat(nth_nat(aa_nat_list_nat(upt(I2),J)),K) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),K) ) ) ).

% nth_upt
tff(fact_5167_upt__0,axiom,
    ! [I2: nat] : aa_nat_list_nat(upt(I2),zero_zero_nat) = nil_nat ).

% upt_0
tff(fact_5168_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,suc),aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N)) ).

% map_Suc_upt
tff(fact_5169_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( cons_nat(M2,cons_nat(N,Ns)) = aa_nat_list_nat(upt(M2),Q2) )
    <=> ( cons_nat(N,Ns) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_5170_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),J) = cons_nat(I2,aa_nat_list_nat(upt(aa_nat_nat(suc,I2)),J)) ) ) ).

% upt_conv_Cons
tff(fact_5171_upt__rec,axiom,
    ! [I2: nat,J: nat] :
      aa_nat_list_nat(upt(I2),J) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J),cons_nat(I2,aa_nat_list_nat(upt(aa_nat_nat(suc,I2)),J)),nil_nat) ).

% upt_rec
tff(fact_5172_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_nat),aa_nat_list_nat(upt(M2),N)) ).

% sorted_wrt_upt
tff(fact_5173_sorted__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_o(sorted_wrt_nat(ord_less_eq_nat),aa_nat_list_nat(upt(M2),N)) ).

% sorted_upt
tff(fact_5174_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( aa_nat_list_nat(upt(I2),J) = cons_nat(X,Xs) )
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
        & ( I2 = X )
        & ( aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),one_one_nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_5175_upto__aux__def,axiom,
    ! [I2: int,J: int,Js: list_int] : upto_aux(I2,J,Js) = append_int(aa_int_list_int(upto(I2),J),Js) ).

% upto_aux_def
tff(fact_5176_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I2),K) = append_int(aa_int_list_int(upto(I2),J),aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int)),K)) ) ) ) ).

% upto_split2
tff(fact_5177_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I2),K) = append_int(aa_int_list_int(upto(I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),aa_int_list_int(upto(J),K)) ) ) ) ).

% upto_split1
tff(fact_5178_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,I2),J)
     => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,J),K)
       => ( aa_int_list_int(upto(I2),K) = append_int(aa_int_list_int(upto(I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int)),cons_int(J,aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,J),one_one_int)),K))) ) ) ) ).

% upto_split3
tff(fact_5179_bezw__0,axiom,
    ! [X: nat] : bezw(X,zero_zero_nat) = product_Pair_int_int(one_one_int,zero_zero_int) ).

% bezw_0
tff(fact_5180_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),aa_nat_nat(suc,J)) = append_nat(aa_nat_list_nat(upt(I2),J),cons_nat(J,nil_nat)) ) ) ).

% upt_Suc_append
tff(fact_5181_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      aa_nat_list_nat(upt(I2),aa_nat_nat(suc,J)) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J),append_nat(aa_nat_list_nat(upt(I2),J),cons_nat(J,nil_nat)),nil_nat) ).

% upt_Suc
tff(fact_5182_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,I2),J)
     => ( aa_nat_list_nat(upt(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K)) = append_nat(aa_nat_list_nat(upt(I2),J),aa_nat_list_nat(upt(J),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,J),K))) ) ) ).

% upt_add_eq_append
tff(fact_5183_bezw_Oelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( Y = $ite(Xa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) ) ) ).

% bezw.elims
tff(fact_5184_bezw_Osimps,axiom,
    ! [X: nat,Y: nat] :
      bezw(X,Y) = $ite(Y = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_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,X),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,X),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,X),Y)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Y)))))) ).

% bezw.simps
tff(fact_5185_nth__upto,axiom,
    ! [I2: 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,I2),aa_nat_int(semiri1314217659103216013at_int,K))),J)
     => ( aa_nat_int(nth_int(aa_int_list_int(upto(I2),J)),K) = aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),aa_nat_int(semiri1314217659103216013at_int,K)) ) ) ).

% nth_upto
tff(fact_5186_int__ops_I2_J,axiom,
    aa_nat_int(semiri1314217659103216013at_int,one_one_nat) = one_one_int ).

% int_ops(2)
tff(fact_5187_int__ops_I4_J,axiom,
    ! [A: nat] : aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(suc,A)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_nat_int(semiri1314217659103216013at_int,A)),one_one_int) ).

% int_ops(4)
tff(fact_5188_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_5189_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_5190_bezw_Opelims,axiom,
    ! [X: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( bezw(X,Xa) = Y )
     => ( aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),product_Pair_nat_nat(X,Xa))
       => ~ ( ( Y = $ite(Xa = zero_zero_nat,product_Pair_int_int(one_one_int,zero_zero_int),product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa))),aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_int_int(aa_int_fun_int_int(times_times_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,bezw(Xa,aa_nat_nat(aa_nat_fun_nat_nat(modulo_modulo_nat,X),Xa)))),aa_nat_int(semiri1314217659103216013at_int,aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,X),Xa)))))) )
           => ~ aa_Pro6150637604445964947_nat_o(accp_P4275260045618599050at_nat(bezw_rel),product_Pair_nat_nat(X,Xa)) ) ) ) ).

% bezw.pelims
tff(fact_5191_take__upt,axiom,
    ! [I2: nat,M2: 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,I2),M2)),N)
     => ( take_nat(M2,aa_nat_list_nat(upt(I2),N)) = aa_nat_list_nat(upt(I2),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)) ) ) ).

% take_upt
tff(fact_5192_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
tff(fact_5193_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
     => ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M2),N)
       => aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,M2),aa_nat_nat(semiri1408675320244567234ct_nat,N)) ) ) ).

% dvd_fact
tff(fact_5194_choose__two,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),numeral_numeral_nat(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))),numeral_numeral_nat(bit0(one))) ).

% choose_two
tff(fact_5195_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_5196_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_5197_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( last_nat(aa_nat_list_nat(upt(I2),J)) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),one_one_nat) ) ) ).

% last_upt
tff(fact_5198_choose__one,axiom,
    ! [N: nat] : aa_nat_nat(aa_nat_fun_nat_nat(binomial,N),one_one_nat) = N ).

% choose_one
tff(fact_5199_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_5200_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_5201_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_5202_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_5203_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,numeral_numeral_nat(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_5204_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_5205_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),aa_nat_nat(suc,I2)))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I2),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
tff(fact_5206_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] : drop_nat(M2,aa_nat_list_nat(upt(I2),J)) = aa_nat_list_nat(upt(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),M2)),J) ).

% drop_upt
tff(fact_5207_greaterThanLessThan__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or5834768355832116004an_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(aa_nat_nat(suc,N)),M2)) ).

% greaterThanLessThan_upt
tff(fact_5208_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(set_or5834768355832116004an_nat(aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanLessThan
tff(fact_5209_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,J),I2))
     => ( aa_nat_nat(nth_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I2),J))),N) = aa_nat_nat(suc,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,I2),N)) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
tff(fact_5210_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_atMost_nat,K)),cons_nat(aa_nat_nat(suc,K),nil_nat)) ).

% sorted_list_of_set_atMost_Suc
tff(fact_5211_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,aa_nat_nat(suc,K))) = append_nat(linord2614967742042102400et_nat(aa_nat_set_nat(set_ord_lessThan_nat,K)),cons_nat(K,nil_nat)) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_5212_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,I2),J)
     => ( hd_nat(aa_nat_list_nat(upt(I2),J)) = I2 ) ) ).

% hd_upt
tff(fact_5213_distinct__upt,axiom,
    ! [I2: nat,J: nat] : aa_list_nat_o(distinct_nat,aa_nat_list_nat(upt(I2),J)) ).

% distinct_upt
tff(fact_5214_distinct__upto,axiom,
    ! [I2: int,J: int] : aa_list_int_o(distinct_int,aa_int_list_int(upto(I2),J)) ).

% distinct_upto
tff(fact_5215_atLeast__upt,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_lessThan_nat,N) = set_nat2(aa_nat_list_nat(upt(zero_zero_nat),N)) ).

% atLeast_upt
tff(fact_5216_greaterThanAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or6659071591806873216st_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(aa_nat_nat(suc,N)),aa_nat_nat(suc,M2))) ).

% greaterThanAtMost_upt
tff(fact_5217_atMost__upto,axiom,
    ! [N: nat] : aa_nat_set_nat(set_ord_atMost_nat,N) = set_nat2(aa_nat_list_nat(upt(zero_zero_nat),aa_nat_nat(suc,N))) ).

% atMost_upto
tff(fact_5218_butlast__upt,axiom,
    ! [M2: nat,N: nat] : butlast_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ).

% butlast_upt
tff(fact_5219_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,aa_nat_nat(suc,I2)),J)
     => ( linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(I2),J)) = cons_nat(aa_nat_nat(suc,I2),linord2614967742042102400et_nat(aa_nat_set_nat(set_or6659071591806873216st_nat(aa_nat_nat(suc,I2)),J))) ) ) ).

% sorted_list_of_set_greaterThanAtMost
tff(fact_5220_greaterThanLessThan__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(set_or5832277885323065728an_int(I2),J) = set_int2(aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% greaterThanLessThan_upto
tff(fact_5221_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(uminus_uminus_rat(numeral_numeral_rat(K))) = product_Pair_int_int(aa_int_int(uminus_uminus_int,numeral_numeral_int(K)),one_one_int) ).

% quotient_of_number(5)
tff(fact_5222_cppi,axiom,
    ! [D: int,Pa: fun_int_o,P: fun_int_o,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,Z4),X2)
           => ( aa_int_o(Pa,X2)
            <=> aa_int_o(P,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                 => ! [Xb2: int] :
                      ( member_int2(Xb2,A2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(Pa,X2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),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(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                    & aa_int_o(P,X4) )
                | ? [X4: int] :
                    ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                    & ? [Xa2: int] :
                        ( member_int2(Xa2,A2)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_5223_card__UNIV__unit,axiom,
    finite410649719033368117t_unit(top_to1996260823553986621t_unit) = one_one_nat ).

% card_UNIV_unit
tff(fact_5224_rat__one__code,axiom,
    quotient_of(one_one_rat) = product_Pair_int_int(one_one_int,one_one_int) ).

% rat_one_code
tff(fact_5225_rat__zero__code,axiom,
    quotient_of(zero_zero_rat) = product_Pair_int_int(zero_zero_int,one_one_int) ).

% rat_zero_code
tff(fact_5226_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(set_or1266510415728281911st_int(L),U)) = 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,U),L)),one_one_int)) ).

% card_atLeastAtMost_int
tff(fact_5227_quotient__of__number_I4_J,axiom,
    quotient_of(uminus_uminus_rat(one_one_rat)) = product_Pair_int_int(aa_int_int(uminus_uminus_int,one_one_int),one_one_int) ).

% quotient_of_number(4)
tff(fact_5228_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(numeral_numeral_rat(K)) = product_Pair_int_int(numeral_numeral_int(K),one_one_int) ).

% quotient_of_number(3)
tff(fact_5229_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : finite_card_int(aa_int_set_int(set_or5832277885323065728an_int(L),U)) = aa_int_nat(nat2,aa_int_int(aa_int_fun_int_int(minus_minus_int,U),aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int))) ).

% card_greaterThanLessThan_int
tff(fact_5230_sgn__rat__def,axiom,
    ! [A: rat] :
      sgn_sgn_rat(A) = $ite(
        A = zero_zero_rat,
        zero_zero_rat,
        $ite(aa_rat_o(ord_less_rat(zero_zero_rat),A),one_one_rat,uminus_uminus_rat(one_one_rat)) ) ).

% sgn_rat_def
tff(fact_5231_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(set_or6656581121297822940st_int(L),U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
tff(fact_5232_atLeastAtMost__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(set_or1266510415728281911st_int(I2),J) = set_int2(aa_int_list_int(upto(I2),J)) ).

% atLeastAtMost_upto
tff(fact_5233_bset_I1_J,axiom,
    ! [D: int,B2: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb2: int] :
                  ( member_int2(Xb2,B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(Pa,X2)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb2: int] :
                    ( member_int2(Xb2,B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Qa,X2)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X3)
                & aa_int_o(Qa,X3) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D))
                & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) ) ) ) ) ) ).

% bset(1)
tff(fact_5234_bset_I2_J,axiom,
    ! [D: int,B2: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb2: int] :
                  ( member_int2(Xb2,B2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(Pa,X2)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb2: int] :
                    ( member_int2(Xb2,B2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Qa,X2)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X3)
                | aa_int_o(Qa,X3) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D))
                | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) ) ) ) ) ) ).

% bset(2)
tff(fact_5235_aset_I1_J,axiom,
    ! [D: int,A2: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb2: int] :
                  ( member_int2(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(Pa,X2)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb2: int] :
                    ( member_int2(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Qa,X2)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X3)
                & aa_int_o(Qa,X3) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D))
                & aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) ) ) ) ) ) ).

% aset(1)
tff(fact_5236_aset_I2_J,axiom,
    ! [D: int,A2: set_int,Pa: fun_int_o,Qa: fun_int_o] :
      ( ! [X2: int] :
          ( ! [Xa3: int] :
              ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb2: int] :
                  ( member_int2(Xb2,A2)
                 => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
         => ( aa_int_o(Pa,X2)
           => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) ) )
     => ( ! [X2: int] :
            ( ! [Xa3: int] :
                ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb2: int] :
                    ( member_int2(Xb2,A2)
                   => ( X2 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb2),Xa3) ) ) )
           => ( aa_int_o(Qa,X2)
             => aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X2),D)) ) )
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( aa_int_o(Pa,X3)
                | aa_int_o(Qa,X3) )
             => ( aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D))
                | aa_int_o(Qa,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) ) ) ) ) ) ).

% aset(2)
tff(fact_5237_greaterThanAtMost__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(set_or6656581121297822940st_int(I2),J) = set_int2(aa_int_list_int(upto(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J)) ).

% greaterThanAtMost_upto
tff(fact_5238_bset_I9_J,axiom,
    ! [D2: int,D: int,B2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)),T3)) ) ) ) ).

% bset(9)
tff(fact_5239_bset_I10_J,axiom,
    ! [D2: int,D: int,B2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)),T3)) ) ) ) ).

% bset(10)
tff(fact_5240_aset_I9_J,axiom,
    ! [D2: int,D: int,A2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
           => aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)),T3)) ) ) ) ).

% aset(9)
tff(fact_5241_aset_I10_J,axiom,
    ! [D2: int,D: int,A2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),T3))
           => ~ aa_int_o(aa_int_fun_int_o(dvd_dvd_int,D2),aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)),T3)) ) ) ) ).

% aset(10)
tff(fact_5242_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N))
     => ( aa_int_set_int(set_or1266510415728281911st_int(M2),aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N)) = insert_int2(aa_int_int(aa_int_fun_int_int(plus_plus_int,one_one_int),N),aa_int_set_int(set_or1266510415728281911st_int(M2),N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_5243_periodic__finite__ex,axiom,
    ! [D2: int,Pa: fun_int_o] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D2)
     => ( ! [X2: int,K2: int] :
            ( aa_int_o(Pa,X2)
          <=> aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),aa_int_int(aa_int_fun_int_int(times_times_int,K2),D2))) )
       => ( ? [X_12: int] : aa_int_o(Pa,X_12)
        <=> ? [X4: int] :
              ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D2))
              & aa_int_o(Pa,X4) ) ) ) ) ).

% periodic_finite_ex
tff(fact_5244_aset_I7_J,axiom,
    ! [D: int,A2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T3),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,T3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) ) ) ) ).

% aset(7)
tff(fact_5245_aset_I5_J,axiom,
    ! [D: int,T3: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(T3,A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)),T3) ) ) ) ) ).

% aset(5)
tff(fact_5246_aset_I4_J,axiom,
    ! [D: int,T3: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(T3,A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X3 != T3 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D) != T3 ) ) ) ) ) ).

% aset(4)
tff(fact_5247_aset_I3_J,axiom,
    ! [D: int,T3: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(aa_int_int(aa_int_fun_int_int(plus_plus_int,T3),one_one_int),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( ( X3 = T3 )
             => ( aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D) = T3 ) ) ) ) ) ).

% aset(3)
tff(fact_5248_bset_I7_J,axiom,
    ! [D: int,T3: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(T3,B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_int,T3),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_int,T3),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) ) ) ) ) ).

% bset(7)
tff(fact_5249_bset_I5_J,axiom,
    ! [D: int,B2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_int,X3),T3)
           => aa_int_o(aa_int_fun_int_o(ord_less_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)),T3) ) ) ) ).

% bset(5)
tff(fact_5250_bset_I4_J,axiom,
    ! [D: int,T3: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(T3,B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X3 != T3 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D) != T3 ) ) ) ) ) ).

% bset(4)
tff(fact_5251_bset_I3_J,axiom,
    ! [D: int,T3: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(aa_int_int(aa_int_fun_int_int(minus_minus_int,T3),one_one_int),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( ( X3 = T3 )
             => ( aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D) = T3 ) ) ) ) ) ).

% bset(3)
tff(fact_5252_rat__sgn__code,axiom,
    ! [P4: rat] : quotient_of(sgn_sgn_rat(P4)) = product_Pair_int_int(aa_int_int(sgn_sgn_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,quotient_of(P4))),one_one_int) ).

% rat_sgn_code
tff(fact_5253_simp__from__to,axiom,
    ! [I2: int,J: int] :
      aa_int_set_int(set_or1266510415728281911st_int(I2),J) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_int,J),I2),bot_bot_set_int,insert_int2(I2,aa_int_set_int(set_or1266510415728281911st_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,I2),one_one_int)),J))) ).

% simp_from_to
tff(fact_5254_aset_I8_J,axiom,
    ! [D: int,A2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,A2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T3),X3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T3),aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)) ) ) ) ).

% aset(8)
tff(fact_5255_aset_I6_J,axiom,
    ! [D: int,T3: int,A2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(aa_int_int(aa_int_fun_int_int(plus_plus_int,T3),one_one_int),A2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,A2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(minus_minus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(plus_plus_int,X3),D)),T3) ) ) ) ) ).

% aset(6)
tff(fact_5256_bset_I8_J,axiom,
    ! [D: int,T3: int,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( member_int2(aa_int_int(aa_int_fun_int_int(minus_minus_int,T3),one_one_int),B2)
       => ! [X3: int] :
            ( ! [Xa4: int] :
                ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
               => ! [Xb3: int] :
                    ( member_int2(Xb3,B2)
                   => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
           => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T3),X3)
             => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,T3),aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)) ) ) ) ) ).

% bset(8)
tff(fact_5257_bset_I6_J,axiom,
    ! [D: int,B2: set_int,T3: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ! [X3: int] :
          ( ! [Xa4: int] :
              ( member_int2(Xa4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
             => ! [Xb3: int] :
                  ( member_int2(Xb3,B2)
                 => ( X3 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb3),Xa4) ) ) )
         => ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,X3),T3)
           => aa_int_o(aa_int_fun_int_o(ord_less_eq_int,aa_int_int(aa_int_fun_int_int(minus_minus_int,X3),D)),T3) ) ) ) ).

% bset(6)
tff(fact_5258_cpmi,axiom,
    ! [D: int,Pa: fun_int_o,P: fun_int_o,B2: set_int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),D)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa_int_o(aa_int_fun_int_o(ord_less_int,X2),Z4)
           => ( aa_int_o(Pa,X2)
            <=> aa_int_o(P,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa3: int] :
                  ( member_int2(Xa3,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                 => ! [Xb2: int] :
                      ( member_int2(Xb2,B2)
                     => ( X2 != aa_int_int(aa_int_fun_int_int(plus_plus_int,Xb2),Xa3) ) ) )
             => ( aa_int_o(Pa,X2)
               => aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(minus_minus_int,X2),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(Pa,X_12)
            <=> ( ? [X4: int] :
                    ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                    & aa_int_o(P,X4) )
                | ? [X4: int] :
                    ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(one_one_int),D))
                    & ? [Xa2: int] :
                        ( member_int2(Xa2,B2)
                        & aa_int_o(Pa,aa_int_int(aa_int_fun_int_int(plus_plus_int,Xa2),X4)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_5259_quotient__of__int,axiom,
    ! [A: int] : quotient_of(of_int(A)) = product_Pair_int_int(A,one_one_int) ).

% quotient_of_int
tff(fact_5260_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(numeral_numeral_int(K),one_one_int)) = numeral_numeral_rat(K) ).

% Frct_code_post(4)
tff(fact_5261_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(product_Pair_int_int(one_one_int,numeral_numeral_int(K))) = aa_rat_rat(aa_rat_fun_rat_rat(divide_divide_rat,one_one_rat),numeral_numeral_rat(K)) ).

% Frct_code_post(5)
tff(fact_5262_all__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
         => aa_nat_o(Pa,M4) )
    <=> ! [X4: nat] :
          ( member_nat2(X4,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less
tff(fact_5263_ex__nat__less,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,M4),N)
          & aa_nat_o(Pa,M4) )
    <=> ? [X4: nat] :
          ( member_nat2(X4,aa_nat_set_nat(set_or1269000886237332187st_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less
tff(fact_5264_atLeastAtMost__upt,axiom,
    ! [N: nat,M2: nat] : aa_nat_set_nat(set_or1269000886237332187st_nat(N),M2) = set_nat2(aa_nat_list_nat(upt(N),aa_nat_nat(suc,M2))) ).

% atLeastAtMost_upt
tff(fact_5265_Frct__code__post_I3_J,axiom,
    frct(product_Pair_int_int(one_one_int,one_one_int)) = one_one_rat ).

% Frct_code_post(3)
tff(fact_5266_zero__rat__def,axiom,
    zero_zero_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(zero_zero_int,one_one_int)) ).

% zero_rat_def
tff(fact_5267_one__rat__def,axiom,
    one_one_rat = aa_Pro6166618844887768661nt_rat(abs_Rat,product_Pair_int_int(one_one_int,one_one_int)) ).

% one_rat_def
tff(fact_5268_zero__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,product_Pair_int_int(zero_zero_int,one_one_int)),zero_zero_rat) ).

% zero_rat.transfer
tff(fact_5269_image__Suc__atMost,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_atMost_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),aa_nat_nat(suc,N)) ).

% image_Suc_atMost
tff(fact_5270_image__Suc__lessThan,axiom,
    ! [N: nat] : image_nat_nat(suc,aa_nat_set_nat(set_ord_lessThan_nat,N)) = aa_nat_set_nat(set_or1269000886237332187st_nat(one_one_nat),N) ).

% image_Suc_lessThan
tff(fact_5271_one__rat_Otransfer,axiom,
    aa_rat_o(aa_Pro2298929076809959388_rat_o(pcr_rat,product_Pair_int_int(one_one_int,one_one_int)),one_one_rat) ).

% one_rat.transfer
tff(fact_5272_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] : linord2614967742042102400et_nat(aa_nat_set_nat(set_or4665077453230672383an_nat(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% sorted_list_of_set_range
tff(fact_5273_ex__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ? [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
          & aa_nat_o(Pa,M4) )
    <=> ? [X4: nat] :
          ( member_nat2(X4,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
          & aa_nat_o(Pa,X4) ) ) ).

% ex_nat_less_eq
tff(fact_5274_all__nat__less__eq,axiom,
    ! [N: nat,Pa: fun_nat_o] :
      ( ! [M4: nat] :
          ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,M4),N)
         => aa_nat_o(Pa,M4) )
    <=> ! [X4: nat] :
          ( member_nat2(X4,aa_nat_set_nat(set_or4665077453230672383an_nat(zero_zero_nat),N))
         => aa_nat_o(Pa,X4) ) ) ).

% all_nat_less_eq
tff(fact_5275_atLeastLessThan__upt,axiom,
    ! [I2: nat,J: nat] : aa_nat_set_nat(set_or4665077453230672383an_nat(I2),J) = set_nat2(aa_nat_list_nat(upt(I2),J)) ).

% atLeastLessThan_upt
tff(fact_5276_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(set_or4662586982721622107an_int(L),aa_int_int(aa_int_fun_int_int(plus_plus_int,U),one_one_int)) = aa_int_set_int(set_or1266510415728281911st_int(L),U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5277_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : aa_int_set_int(set_or4662586982721622107an_int(aa_int_int(aa_int_fun_int_int(plus_plus_int,L),one_one_int)),U) = aa_int_set_int(set_or5832277885323065728an_int(L),U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_5278_atLeastLessThan__upto,axiom,
    ! [I2: int,J: int] : aa_int_set_int(set_or4662586982721622107an_int(I2),J) = set_int2(aa_int_list_int(upto(I2),aa_int_int(aa_int_fun_int_int(minus_minus_int,J),one_one_int))) ).

% atLeastLessThan_upto
tff(fact_5279_bool__assn__proper_I4_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Qa)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(Pa),Qa)) ) ) ).

% bool_assn_proper(4)
tff(fact_5280_bool__assn__proper_I3_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o,Qa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => ( aa_fun1675921005893230629at_o_o(proper,Qa)
       => aa_fun1675921005893230629at_o_o(proper,aa_fun2638059832795814268_nat_o(aTP_Lamm_af(Pa),Qa)) ) ) ).

% bool_assn_proper(3)
tff(fact_5281_bool__assn__proper_I2_J,axiom,
    aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_ag) ).

% bool_assn_proper(2)
tff(fact_5282_tl__upt,axiom,
    ! [M2: nat,N: nat] : tl_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),N) ).

% tl_upt
tff(fact_5283_bool__assn__proper_I5_J,axiom,
    ! [Pa: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(proper,Pa)
     => aa_fun1675921005893230629at_o_o(proper,aTP_Lamm_ah(Pa)) ) ).

% bool_assn_proper(5)
tff(fact_5284_bot__assn__def,axiom,
    bot_bot_assn = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_ag) ).

% bot_assn_def
tff(fact_5285_sup__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(sup_sup_assn,Pa),Qa) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(Pa),Qa)) ).

% sup_assn_def
tff(fact_5286_inf__assn__def,axiom,
    ! [Pa: assn,Qa: assn] : aa_assn_assn(aa_ass954072293579886259n_assn(inf_inf_assn,Pa),Qa) = aa_fun5526454687433275355o_assn(abs_assn,aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(Pa),Qa)) ).

% inf_assn_def
tff(fact_5287_map__add__upt,axiom,
    ! [N: nat,M2: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,aa_nat_fun_nat_nat(aTP_Lamm_ak,N)),aa_nat_list_nat(upt(zero_zero_nat),M2)) = aa_nat_list_nat(upt(N),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,M2),N)) ).

% map_add_upt
tff(fact_5288_uminus__assn__def,axiom,
    ! [Pa: assn] : aa_assn_assn(uminus_uminus_assn,Pa) = aa_fun5526454687433275355o_assn(abs_assn,aTP_Lamm_al(Pa)) ).

% uminus_assn_def
tff(fact_5289_map__decr__upt,axiom,
    ! [M2: nat,N: nat] : aa_list_nat_list_nat(aa_fun4736978287202700568st_nat(map_nat_nat,aTP_Lamm_am),aa_nat_list_nat(upt(aa_nat_nat(suc,M2)),aa_nat_nat(suc,N))) = aa_nat_list_nat(upt(M2),N) ).

% map_decr_upt
tff(fact_5290_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),M2)
     => ( aa_nat_nat(aa_nat_fun_nat_nat(divide_divide_nat,aa_nat_nat(semiri1408675320244567234ct_nat,M2)),aa_nat_nat(semiri1408675320244567234ct_nat,N)) = groups708209901874060359at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)),M2)) ) ) ).

% fact_div_fact
tff(fact_5291_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,N),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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),M2)),one_one_nat)),M2) ).

% choose_rising_sum(2)
tff(fact_5292_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,N),aa_nat_set_nat(set_ord_atMost_nat,M2)) = 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),M2)),one_one_nat)),aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N),one_one_nat)) ).

% choose_rising_sum(1)
tff(fact_5293_sum__power2,axiom,
    ! [K: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_set_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,numeral_numeral_nat(bit0(one))),K)),one_one_nat) ).

% sum_power2
tff(fact_5294_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] : groups3542108847815614940at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or4665077453230672383an_nat(M2),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,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Ico_nat
tff(fact_5295_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] : groups3542108847815614940at_nat(aTP_Lamm_an,aa_nat_set_nat(set_or1269000886237332187st_nat(M2),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,M2),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,M2),one_one_nat)))),numeral_numeral_nat(bit0(one))) ).

% Sum_Icc_nat
tff(fact_5296_binomial__r__part__sum,axiom,
    ! [M2: 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,numeral_numeral_nat(bit0(one))),M2)),one_one_nat)),aa_nat_set_nat(set_ord_atMost_nat,M2)) = aa_nat_nat(aa_nat_fun_nat_nat(power_power_nat,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),M2)) ).

% binomial_r_part_sum
tff(fact_5297_choose__linear__sum,axiom,
    ! [N: nat] : groups3542108847815614940at_nat(aa_nat_fun_nat_nat(aTP_Lamm_ap,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,numeral_numeral_nat(bit0(one))),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat))) ).

% choose_linear_sum
tff(fact_5298_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod_nat_nat] : unique5026877609467782581ep_nat(L,Qr) = produc2626176000494625587at_nat(aTP_Lamm_aq(L),Qr) ).

% divmod_step_nat_def
tff(fact_5299_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod_int_int] : unique5024387138958732305ep_int(L,Qr) = produc4245557441103728435nt_int(aTP_Lamm_ar(L),Qr) ).

% divmod_step_int_def
tff(fact_5300_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_eq_int,M2),N)
     => ( groups4538972089207619220nt_int(aTP_Lamm_as,aa_int_set_int(set_or1266510415728281911st_int(M2),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,M2),aa_int_int(aa_int_fun_int_int(minus_minus_int,M2),one_one_int)))),numeral_numeral_int(bit0(one))) ) ) ).

% Sum_Icc_int
tff(fact_5301_normalize__def,axiom,
    ! [P4: product_prod_int_int] :
      normalize(P4) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),aa_Pro6799258434465214121nt_int(product_snd_int_int,P4)),
        $let(
          a: int,
          a:= aa_int_int(aa_int_fun_int_int(gcd_gcd_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P4)),
          product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P4)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P4)),a)) ),
        $ite(
          aa_Pro6799258434465214121nt_int(product_snd_int_int,P4) = zero_zero_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,P4)),aa_Pro6799258434465214121nt_int(product_snd_int_int,P4))),
            product_Pair_int_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_fst_int_int,P4)),a),aa_int_int(aa_int_fun_int_int(divide_divide_int,aa_Pro6799258434465214121nt_int(product_snd_int_int,P4)),a)) ) ) ) ).

% normalize_def
tff(fact_5302_remdups__upt,axiom,
    ! [M2: nat,N: nat] : remdups_nat(aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% remdups_upt
tff(fact_5303_gcd__1__int,axiom,
    ! [M2: int] : aa_int_int(aa_int_fun_int_int(gcd_gcd_int,M2),one_one_int) = one_one_int ).

% gcd_1_int
tff(fact_5304_gcd__1__nat,axiom,
    ! [M2: nat] : aa_nat_nat(aa_nat_fun_nat_nat(gcd_gcd_nat,M2),one_one_nat) = one_one_nat ).

% gcd_1_nat
tff(fact_5305_sort__upt,axiom,
    ! [M2: nat,N: nat] : linord738340561235409698at_nat(aTP_Lamm_an,aa_nat_list_nat(upt(M2),N)) = aa_nat_list_nat(upt(M2),N) ).

% sort_upt
tff(fact_5306_sort__upto,axiom,
    ! [I2: int,J: int] : linord1735203802627413978nt_int(aTP_Lamm_as,aa_int_list_int(upto(I2),J)) = aa_int_list_int(upto(I2),J) ).

% sort_upto
tff(fact_5307_rat__inverse__code,axiom,
    ! [P4: rat] : quotient_of(aa_rat_rat(inverse_inverse_rat,P4)) = produc4245557441103728435nt_int(aTP_Lamm_at,quotient_of(P4)) ).

% rat_inverse_code
tff(fact_5308_inverse__rat_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,X),X)
     => ( aa_rat_rat(inverse_inverse_rat,aa_Pro6166618844887768661nt_rat(abs_Rat,X)) = aa_Pro6166618844887768661nt_rat(abs_Rat,
            $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,X) = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,X),aa_Pro6799258434465214121nt_int(product_fst_int_int,X)))) ) ) ).

% inverse_rat.abs_eq
tff(fact_5309_divmod__step__integer__def,axiom,
    ! [L: num,Qr: produc8923325533196201883nteger] : unique4921790084139445826nteger(L,Qr) = produc6916734918728496179nteger(aTP_Lamm_au(L),Qr) ).

% divmod_step_integer_def
tff(fact_5310_inverse__rat_Otransfer,axiom,
    bNF_re8279943556446156061nt_rat(pcr_rat,pcr_rat,aTP_Lamm_av,inverse_inverse_rat) ).

% inverse_rat.transfer
tff(fact_5311_atLeastPlusOneLessThan__greaterThanLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or4266950643985792945nteger(L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_integer
tff(fact_5312_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or8404916559141939852nteger(L,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,U),one_one_Code_integer)) = set_or189985376899183464nteger(L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5313_atLeastPlusOneAtMost__greaterThanAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or189985376899183464nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,L),one_one_Code_integer),U) = set_or2715278749043346189nteger(L,U) ).

% atLeastPlusOneAtMost_greaterThanAtMost_integer
tff(fact_5314_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(ord_le6747313008572928689nteger(K),zero_z3403309356797280102nteger),aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,one_one_Code_integer),one_one_Code_integer) ) ).

% sgn_integer_code
tff(fact_5315_one__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,product_Pair_int_int(one_one_int,one_one_int)),product_Pair_int_int(one_one_int,one_one_int)) ).

% one_rat.rsp
tff(fact_5316_zero__rat_Orsp,axiom,
    aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,product_Pair_int_int(zero_zero_int,one_one_int)),product_Pair_int_int(zero_zero_int,one_one_int)) ).

% zero_rat.rsp
tff(fact_5317_one__integer_Orsp,axiom,
    one_one_int = one_one_int ).

% one_integer.rsp
tff(fact_5318_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
tff(fact_5319_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa_int_o(aa_int_fun_int_o(ord_less_int,K),zero_zero_int),
        aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,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(bit0(one))),code_integer_of_int(aa_int_int(aa_int_fun_int_int(divide_divide_int,K),numeral_numeral_int(bit0(one))))),
            $ite(aa_int_int(aa_int_fun_int_int(modulo_modulo_int,K),numeral_numeral_int(bit0(one))) = zero_zero_int,l,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,l),one_one_Code_integer)) ) ) ) ).

% integer_of_int_code
tff(fact_5320_integer__of__num_I3_J,axiom,
    ! [N: num] :
      code_integer_of_num(bit1(N)) = $let(
        k: code_integer,
        k:= code_integer_of_num(N),
        aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,k),k)),one_one_Code_integer) ) ).

% integer_of_num(3)
tff(fact_5321_Fract_Orsp,axiom,
    bNF_re4808202430149235895nt_int(fequal_int,bNF_re6250860962936578807nt_int(fequal_int,ratrel),aTP_Lamm_aw,aTP_Lamm_aw) ).

% Fract.rsp
tff(fact_5322_one__integer__def,axiom,
    one_one_Code_integer = code_integer_of_int(one_one_int) ).

% one_integer_def
tff(fact_5323_integer__of__num__triv_I1_J,axiom,
    code_integer_of_num(one) = one_one_Code_integer ).

% integer_of_num_triv(1)
tff(fact_5324_FI__def,axiom,
    ! [M2: list_P8527749157015355191n_assn,P4: assn,Q2: assn,Up: assn,Uq: assn,F: assn] :
      ( aa_assn_o(fi(M2,P4,Q2,Up,Uq),F)
    <=> ( ! [X4: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X4,set_Pr1139785259514867910n_assn(M2))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X4) )
       => 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,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),M2),one_one_assn)),P4)),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,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),M2),one_one_assn)),Q2)),Uq)),F)) ) ) ).

% FI_def
tff(fact_5325_inverse__rat_Orsp,axiom,
    bNF_re7145576690424134365nt_int(ratrel,ratrel,aTP_Lamm_av,aTP_Lamm_av) ).

% inverse_rat.rsp
tff(fact_5326_FI__RESULT__def,axiom,
    ! [M: list_P8527749157015355191n_assn,UP: assn,UQ: assn,Fa: assn] :
      ( aa_assn_o(fI_RESULT(M,UP,UQ),Fa)
    <=> ( ! [X4: produc6575502325842934193n_assn] :
            ( member7957490590177025114n_assn(X4,set_Pr1139785259514867910n_assn(M))
           => aa_Pro2491721552408928739assn_o(produc7274209992780475162assn_o(entails),X4) )
       => aa_assn_o(aa_assn_fun_assn_o(entails,aa_assn_assn(aa_ass954072293579886259n_assn(times_times_assn,foldr_assn_assn(times_times_assn,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc9167289414957590229n_assn),M),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,aa_lis6404667907582590259t_assn(aa_fun5966445080204517586t_assn(map_Pr8991440229025900053n_assn,produc2051961928117032727n_assn),M),one_one_assn)),UQ)),Fa)) ) ) ).

% FI_RESULT_def
tff(fact_5327_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa_Code_integer_o(ord_le3102999989581377725nteger(K),one_one_Code_integer),one,produc7336495610019696514er_num(aTP_Lamm_ax,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ).

% num_of_integer_code
tff(fact_5328_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa_Code_integer_o(ord_le6747313008572928689nteger(K),zero_z3403309356797280102nteger),
        aa_int_int(uminus_uminus_int,code_int_of_integer(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,K))),
        $ite(K = zero_z3403309356797280102nteger,zero_zero_int,produc1553301316500091796er_int(aTP_Lamm_ay,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ) ).

% int_of_integer_code
tff(fact_5329_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_az(K),code_divmod_abs(K,numera6620942414471956472nteger(bit0(one))))) ).

% bit_cut_integer_code
tff(fact_5330_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      code_nat_of_integer(K) = $ite(aa_Code_integer_o(ord_le3102999989581377725nteger(K),zero_z3403309356797280102nteger),zero_zero_nat,produc1555791787009142072er_nat(aTP_Lamm_ba,code_divmod_integer(K,numera6620942414471956472nteger(bit0(one))))) ).

% nat_of_integer_code
tff(fact_5331_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one_Code_integer) = one_one_int ).

% one_integer.rep_eq
tff(fact_5332_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(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),L),
          $ite(aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),K),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_bb(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(ord_le6747313008572928689nteger(K),zero_z3403309356797280102nteger),code_divmod_abs(K,L),produc6916734918728496179nteger(aTP_Lamm_bc(L),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_5333_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_bd(L),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_5334_card__length__sum__list,axiom,
    ! [M2: nat,N2: nat] : finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_be(M2),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(binomial,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,N2),M2)),one_one_nat)),N2) ).

% card_length_sum_list
tff(fact_5335_card__length__sum__list__rec,axiom,
    ! [M2: nat,N2: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,one_one_nat),M2)
     => ( finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_be(M2),N2))) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bf(M2),N2)))),finite_card_list_nat(collect_list_nat(aa_nat5883842100386203267_nat_o(aTP_Lamm_bg(M2),N2)))) ) ) ).

% card_length_sum_list_rec
tff(fact_5336_rat__number__expand_I5_J,axiom,
    ! [K: num] : uminus_uminus_rat(numeral_numeral_rat(K)) = aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,numeral_numeral_int(K))),one_one_int) ).

% rat_number_expand(5)
tff(fact_5337_Fract__of__int__eq,axiom,
    ! [K: int] : aa_int_rat(aa_int_fun_int_rat(fract,K),one_one_int) = ring_1_of_int_rat(K) ).

% Fract_of_int_eq
tff(fact_5338_eq__rat_I2_J,axiom,
    ! [A: int] : aa_int_rat(aa_int_fun_int_rat(fract,A),zero_zero_int) = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% eq_rat(2)
tff(fact_5339_Fract__of__nat__eq,axiom,
    ! [K: nat] : aa_int_rat(aa_int_fun_int_rat(fract,aa_nat_int(semiri1314217659103216013at_int,K)),one_one_int) = semiri681578069525770553at_rat(K) ).

% Fract_of_nat_eq
tff(fact_5340_One__rat__def,axiom,
    one_one_rat = aa_int_rat(aa_int_fun_int_rat(fract,one_one_int),one_one_int) ).

% One_rat_def
tff(fact_5341_Zero__rat__def,axiom,
    zero_zero_rat = aa_int_rat(aa_int_fun_int_rat(fract,zero_zero_int),one_one_int) ).

% Zero_rat_def
tff(fact_5342_Fract__add__one,axiom,
    ! [N: int,M2: int] :
      ( ( N != zero_zero_int )
     => ( aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(aa_int_fun_int_int(plus_plus_int,M2),N)),N) = aa_rat_rat(aa_rat_fun_rat_rat(plus_plus_rat,aa_int_rat(aa_int_fun_int_rat(fract,M2),N)),one_one_rat) ) ) ).

% Fract_add_one
tff(fact_5343_rat__number__collapse_I3_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,numeral_numeral_int(W)),one_one_int) = numeral_numeral_rat(W) ).

% rat_number_collapse(3)
tff(fact_5344_rat__number__expand_I3_J,axiom,
    ! [K: num] : numeral_numeral_rat(K) = aa_int_rat(aa_int_fun_int_rat(fract,numeral_numeral_int(K)),one_one_int) ).

% rat_number_expand(3)
tff(fact_5345_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(ord_less_eq_rat(one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,B),A) ) ) ).

% one_le_Fract_iff
tff(fact_5346_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(ord_less_eq_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_eq_int,A),B) ) ) ).

% Fract_le_one_iff
tff(fact_5347_Fract_Oabs__eq,axiom,
    ! [Xa: int,X: int] :
      aa_int_rat(aa_int_fun_int_rat(fract,Xa),X) = aa_Pro6166618844887768661nt_rat(abs_Rat,
        $ite(X = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Xa,X))) ).

% Fract.abs_eq
tff(fact_5348_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( aa_rat_o(ord_less_eq_rat(ring_1_of_int_rat(aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B))),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      & aa_rat_o(ord_less_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),ring_1_of_int_rat(aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(aa_int_fun_int_int(divide_divide_int,A),B)),one_one_int))) ) ).

% rat_floor_lemma
tff(fact_5349_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(ord_less_rat(one_one_rat),aa_int_rat(aa_int_fun_int_rat(fract,A),B))
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,B),A) ) ) ).

% one_less_Fract_iff
tff(fact_5350_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( aa_int_o(aa_int_fun_int_o(ord_less_int,zero_zero_int),B)
     => ( aa_rat_o(ord_less_rat(aa_int_rat(aa_int_fun_int_rat(fract,A),B)),one_one_rat)
      <=> aa_int_o(aa_int_fun_int_o(ord_less_int,A),B) ) ) ).

% Fract_less_one_iff
tff(fact_5351_rat__number__collapse_I5_J,axiom,
    aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,one_one_int)),one_one_int) = uminus_uminus_rat(one_one_rat) ).

% rat_number_collapse(5)
tff(fact_5352_Fract_Otransfer,axiom,
    bNF_re2486238095283335024nt_rat(fequal_int,bNF_re2214769303045360666nt_rat(fequal_int,pcr_rat),aTP_Lamm_aw,fract) ).

% Fract.transfer
tff(fact_5353_rat__number__collapse_I4_J,axiom,
    ! [W: num] : aa_int_rat(aa_int_fun_int_rat(fract,aa_int_int(uminus_uminus_int,numeral_numeral_int(W))),one_one_int) = uminus_uminus_rat(numeral_numeral_rat(W)) ).

% rat_number_collapse(4)
tff(fact_5354_floor__rat__def,axiom,
    ! [X: rat] : archim3151403230148437115or_rat(X) = the_int(aTP_Lamm_bh(X)) ).

% floor_rat_def
tff(fact_5355_abs__division__segment,axiom,
    ! [K: int] : aa_int_int(abs_abs_int,aa_int_int(euclid3395696857347342551nt_int,K)) = one_one_int ).

% abs_division_segment
tff(fact_5356_division__segment__nat__def,axiom,
    ! [N: nat] : aa_nat_nat(euclid3398187327856392827nt_nat,N) = one_one_nat ).

% division_segment_nat_def
tff(fact_5357_division__segment__int__def,axiom,
    ! [K: int] :
      aa_int_int(euclid3395696857347342551nt_int,K) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,zero_zero_int),K),one_one_int,aa_int_int(uminus_uminus_int,one_one_int)) ).

% division_segment_int_def
tff(fact_5358_inverse__rat__def,axiom,
    inverse_inverse_rat = map_fu5673905371560938248nt_rat(rep_Rat,abs_Rat,aTP_Lamm_av) ).

% inverse_rat_def
tff(fact_5359_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,N),one_one_nat)
     => ( num_of_nat(N) = one ) ) ).

% num_of_nat_One
tff(fact_5360_one__int_Otransfer,axiom,
    aa_int_o(pcr_int(product_Pair_nat_nat(one_one_nat,zero_zero_nat)),one_one_int) ).

% one_int.transfer
tff(fact_5361_one__int__def,axiom,
    one_one_int = abs_Integ(product_Pair_nat_nat(one_one_nat,zero_zero_nat)) ).

% one_int_def
tff(fact_5362_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one_nat) = one_one_Code_integer ).

% integer_of_nat_1
tff(fact_5363_list__all__iff__all__interval__int,axiom,
    ! [Pa: fun_int_o,I2: int,J: int] :
      ( aa_list_int_o(list_all_int(Pa),aa_int_list_int(upto(I2),J))
    <=> aa_int_o(aa_int_fun_int_o(all_interval_int(Pa),I2),J) ) ).

% list_all_iff_all_interval_int
tff(fact_5364_list__ex__iff__not__all__inverval__int,axiom,
    ! [Pa: fun_int_o,I2: int,J: int] :
      ( aa_list_int_o(list_ex_int(Pa),aa_int_list_int(upto(I2),J))
    <=> ~ aa_int_o(aa_int_fun_int_o(all_interval_int(comp_o_o_int(fNot,Pa)),I2),J) ) ).

% list_ex_iff_not_all_inverval_int
tff(fact_5365_all__interval__int__def,axiom,
    ! [Pa: fun_int_o,I2: int,J: int] :
      ( aa_int_o(aa_int_fun_int_o(all_interval_int(Pa),I2),J)
    <=> ! [X4: int] :
          ( member_int2(X4,aa_int_set_int(set_or1266510415728281911st_int(I2),J))
         => aa_int_o(Pa,X4) ) ) ).

% all_interval_int_def
tff(fact_5366_unit__factor__simps_I2_J,axiom,
    ! [N: nat] : aa_nat_nat(unit_f2748546683901255202or_nat,aa_nat_nat(suc,N)) = one_one_nat ).

% unit_factor_simps(2)
tff(fact_5367_unit__factor__nat__def,axiom,
    ! [N: nat] :
      aa_nat_nat(unit_f2748546683901255202or_nat,N) = $ite(N = zero_zero_nat,zero_zero_nat,one_one_nat) ).

% unit_factor_nat_def
tff(fact_5368_list__all__iff__all__interval__nat,axiom,
    ! [Pa: fun_nat_o,I2: nat,J: nat] :
      ( aa_list_nat_o(list_all_nat(Pa),aa_nat_list_nat(upt(I2),J))
    <=> aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(Pa),I2),J) ) ).

% list_all_iff_all_interval_nat
tff(fact_5369_list__ex__iff__not__all__inverval__nat,axiom,
    ! [Pa: fun_nat_o,I2: nat,J: nat] :
      ( aa_list_nat_o(list_ex_nat(Pa),aa_nat_list_nat(upt(I2),J))
    <=> ~ aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(comp_o_o_nat(fNot,Pa)),I2),J) ) ).

% list_ex_iff_not_all_inverval_nat
tff(fact_5370_Gcd__nat__eq__one,axiom,
    ! [N2: set_nat] :
      ( member_nat2(one_one_nat,N2)
     => ( gcd_Gcd_nat(N2) = one_one_nat ) ) ).

% Gcd_nat_eq_one
tff(fact_5371_Lcm__nat__empty,axiom,
    gcd_Lcm_nat(bot_bot_set_nat) = one_one_nat ).

% Lcm_nat_empty
tff(fact_5372_all__interval__nat__def,axiom,
    ! [Pa: fun_nat_o,I2: nat,J: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(all_interval_nat(Pa),I2),J)
    <=> ! [X4: nat] :
          ( member_nat2(X4,aa_nat_set_nat(set_or4665077453230672383an_nat(I2),J))
         => aa_nat_o(Pa,X4) ) ) ).

% all_interval_nat_def
tff(fact_5373_lcm__1__iff__int,axiom,
    ! [M2: int,N: int] :
      ( ( aa_int_int(aa_int_fun_int_int(gcd_lcm_int,M2),N) = one_one_int )
    <=> ( ( ( M2 = one_one_int )
          | ( M2 = aa_int_int(uminus_uminus_int,one_one_int) ) )
        & ( ( N = one_one_int )
          | ( N = aa_int_int(uminus_uminus_int,one_one_int) ) ) ) ) ).

% lcm_1_iff_int
tff(fact_5374_Lcm__nat__set__eq__fold,axiom,
    ! [Xs: list_nat] : gcd_Lcm_nat(set_nat2(Xs)) = aa_nat_nat(fold_nat_nat(gcd_lcm_nat,Xs),one_one_nat) ).

% Lcm_nat_set_eq_fold
tff(fact_5375_Lcm__int__set__eq__fold,axiom,
    ! [Xs: list_int] : gcd_Lcm_int(set_int2(Xs)) = aa_int_int(fold_int_int(gcd_lcm_int,Xs),one_one_int) ).

% Lcm_int_set_eq_fold
tff(fact_5376_Lcm__nat__def,axiom,
    ! [M: set_nat] :
      gcd_Lcm_nat(M) = $ite(finite_finite_nat(M),lattic7826324295020591184_F_nat(gcd_lcm_nat,one_one_nat,M),zero_zero_nat) ).

% Lcm_nat_def
tff(fact_5377_coprime__diff__one__right__nat,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(algebr934650988132801477me_nat,N),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)) ) ).

% coprime_diff_one_right_nat
tff(fact_5378_coprime__common__divisor__nat,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(algebr934650988132801477me_nat,A),B)
     => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),A)
       => ( aa_nat_o(aa_nat_fun_nat_o(dvd_dvd_nat,X),B)
         => ( X = one_one_nat ) ) ) ) ).

% coprime_common_divisor_nat
tff(fact_5379_coprime__diff__one__left__nat,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(algebr934650988132801477me_nat,aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,N),one_one_nat)),N) ) ).

% coprime_diff_one_left_nat
tff(fact_5380_in__range_Osimps,axiom,
    ! [H2: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,produc7507926704131184380et_nat(H2,As2))
    <=> ! [X4: nat] :
          ( member_nat2(X4,As2)
         => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H2)) ) ) ).

% in_range.simps
tff(fact_5381_coprime__common__divisor__int,axiom,
    ! [A: int,B: int,X: int] :
      ( aa_int_o(aa_int_fun_int_o(algebr932160517623751201me_int,A),B)
     => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),A)
       => ( aa_int_o(aa_int_fun_int_o(dvd_dvd_int,X),B)
         => ( aa_int_int(abs_abs_int,X) = one_one_int ) ) ) ) ).

% coprime_common_divisor_int
tff(fact_5382_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ! [X2: nat] :
                ( member_nat2(X2,As)
               => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H3)) ) ) ) ).

% in_range.elims(3)
tff(fact_5383_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ~ ! [X3: nat] :
                  ( member_nat2(X3,As)
                 => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H3)) ) ) ) ).

% in_range.elims(2)
tff(fact_5384_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X = produc7507926704131184380et_nat(H3,As) )
           => ( (Y)
            <=> ~ ! [X4: nat] :
                    ( member_nat2(X4,As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H3)) ) ) ) ) ).

% in_range.elims(1)
tff(fact_5385_one__int_Orsp,axiom,
    aa_Pro6150637604445964947_nat_o(intrel(product_Pair_nat_nat(one_one_nat,zero_zero_nat)),product_Pair_nat_nat(one_one_nat,zero_zero_nat)) ).

% one_int.rsp
tff(fact_5386_wait__rule,axiom,
    ! [N: nat] : hoare_8945653483474564448t_unit(one_one_assn,heap_Time_wait(N),aTP_Lamm_bi) ).

% wait_rule
tff(fact_5387_inf__unit__def,axiom,
    ! [Uu: product_unit,Uv: product_unit] : inf_inf_Product_unit(Uu,Uv) = product_Unity ).

% inf_unit_def
tff(fact_5388_assert_H__rule,axiom,
    ! [Pa: assn,Phi: $o] :
      ( ! [H3: produc3658429121746597890et_nat] :
          ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Pa),H3)
         => (Phi) )
     => hoare_8945653483474564448t_unit(Pa,refine_Imp_assert((Phi)),aTP_Lamm_bj(Pa)) ) ).

% assert'_rule
tff(fact_5389_natural__zero__minus__one,axiom,
    minus_7197305767214868737atural(zero_z2226904508553997617atural,one_one_Code_natural) = zero_z2226904508553997617atural ).

% natural_zero_minus_one
tff(fact_5390_log_Oelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( Y = $ite(
            ( aa_Code_natural_o(ord_le1926595141338095240atural(X),one_one_Code_natural)
            | aa_Code_natural_o(ord_le5570908160329646204atural(Xa),X) ),
            one_one_Code_natural,
            aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa,X))) ) ) ) ).

% log.elims
tff(fact_5391_log_Osimps,axiom,
    ! [B: code_natural,I2: code_natural] :
      log(B,I2) = $ite(
        ( aa_Code_natural_o(ord_le1926595141338095240atural(B),one_one_Code_natural)
        | aa_Code_natural_o(ord_le5570908160329646204atural(I2),B) ),
        one_one_Code_natural,
        aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(B,divide5121882707175180666atural(I2,B))) ) ).

% log.simps
tff(fact_5392_next_Osimps,axiom,
    ! [V2: code_natural,W: code_natural] :
      aa_Pro7165575656943818355atural(next,produc3574140220909816553atural(V2,W)) = $let(
        v: code_natural,
        v:= minus_shift(numera5444537566228673987atural(bit1(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(V2,numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit0(bit1(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(V2,numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit0(bit1(bit0(bit1(bit1(bit0(bit0(bit0(bit1(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit0(one)))))))))))))))),
        $let(
          w: code_natural,
          w:= minus_shift(numera5444537566228673987atural(bit1(bit1(bit1(bit0(bit0(bit0(bit0(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,modulo8411746178871703098atural(W,numera5444537566228673987atural(bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit0(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit1(bit1(bit1(bit1(bit0(bit0(one))))))))))))))))),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,divide5121882707175180666atural(W,numera5444537566228673987atural(bit0(bit1(bit1(bit0(bit0(bit1(bit0(bit0(bit0(bit1(bit1(bit1(bit0(bit0(bit1(one)))))))))))))))))),numera5444537566228673987atural(bit1(bit1(bit1(bit1(bit0(bit0(bit1(bit1(bit0(bit1(bit1(one)))))))))))))),
          aa_Pro7165575656943818355atural(produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,minus_shift(numera5444537566228673987atural(bit0(bit1(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),v,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,w),one_one_Code_natural))),one_one_Code_natural)),produc3574140220909816553atural(v,w)) ) ) ).

% next.simps
tff(fact_5393_Random_Orange__def,axiom,
    ! [K: code_natural] : range(K) = produc5538323210962509403atural(iterat8892046348760725948atural(log(numera5444537566228673987atural(bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one))))))))))))))))))))))))))))))),K),aTP_Lamm_bl,one_one_Code_natural),aTP_Lamm_bm(K)) ).

% Random.range_def
tff(fact_5394_inc__shift__def,axiom,
    ! [V2: code_natural,K: code_natural] :
      inc_shift(V2,K) = $ite(V2 = K,one_one_Code_natural,aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,K),one_one_Code_natural)) ).

% inc_shift_def
tff(fact_5395_log_Opelims,axiom,
    ! [X: code_natural,Xa: code_natural,Y: code_natural] :
      ( ( log(X,Xa) = Y )
     => ( accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa))
       => ~ ( ( Y = $ite(
                  ( aa_Code_natural_o(ord_le1926595141338095240atural(X),one_one_Code_natural)
                  | aa_Code_natural_o(ord_le5570908160329646204atural(Xa),X) ),
                  one_one_Code_natural,
                  aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,one_one_Code_natural),log(X,divide5121882707175180666atural(Xa,X))) ) )
           => ~ accp_P8126237942716283194atural(log_rel,produc3574140220909816553atural(X,Xa)) ) ) ) ).

% log.pelims
tff(fact_5396_one__natural__def,axiom,
    one_one_Code_natural = code_natural_of_nat(one_one_nat) ).

% one_natural_def
tff(fact_5397_Suc__natural__minus__one,axiom,
    ! [N: code_natural] : minus_7197305767214868737atural(code_Suc(N),one_one_Code_natural) = N ).

% Suc_natural_minus_one
tff(fact_5398_Code__Numeral_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] : code_sub(bit0(M2),bit1(N)) = minus_8373710615458151222nteger(code_dup(code_sub(M2,N)),one_one_Code_integer) ).

% Code_Numeral.sub_code(9)
tff(fact_5399_Code__Numeral_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] : code_sub(bit1(M2),bit0(N)) = aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,code_dup(code_sub(M2,N))),one_one_Code_integer) ).

% Code_Numeral.sub_code(8)
tff(fact_5400_Int_Osub__code_I9_J,axiom,
    ! [M2: num,N: num] : sub(bit0(M2),bit1(N)) = aa_int_int(aa_int_fun_int_int(minus_minus_int,aa_int_int(dup,sub(M2,N))),one_one_int) ).

% Int.sub_code(9)
tff(fact_5401_Int_Osub__code_I8_J,axiom,
    ! [M2: num,N: num] : sub(bit1(M2),bit0(N)) = aa_int_int(aa_int_fun_int_int(plus_plus_int,aa_int_int(dup,sub(M2,N))),one_one_int) ).

% Int.sub_code(8)
tff(fact_5402_Fract__def,axiom,
    fract = map_fu7945985449245211511nt_rat(id_int,map_fu3424225382358772769nt_rat(id_int,abs_Rat),aTP_Lamm_aw) ).

% Fract_def
tff(fact_5403_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As))
               => ! [X2: nat] :
                    ( member_nat2(X2,As)
                   => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X2),lim_Product_unit(H3)) ) ) ) ) ) ).

% in_range.pelims(3)
tff(fact_5404_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(in_range,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( ( (Y)
                <=> ! [X4: nat] :
                      ( member_nat2(X4,As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X4),lim_Product_unit(H3)) ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As)) ) ) ) ) ).

% in_range.pelims(1)
tff(fact_5405_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(in_range,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(in_range_rel),produc7507926704131184380et_nat(H3,As))
               => ~ ! [X3: nat] :
                      ( member_nat2(X3,As)
                     => aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,X3),lim_Product_unit(H3)) ) ) ) ) ) ).

% in_range.pelims(2)
tff(fact_5406_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As))
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
tff(fact_5407_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As))
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
tff(fact_5408_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( aa_Pro653082453696054616_nat_o(one_assn_raw,X)
      <=> (Y) )
     => ( aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),X)
       => ~ ! [H3: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X = produc7507926704131184380et_nat(H3,As) )
             => ( ( (Y)
                <=> ( As = bot_bot_set_nat ) )
               => ~ aa_Pro653082453696054616_nat_o(accp_P5801069581201407417et_nat(one_assn_raw_rel),produc7507926704131184380et_nat(H3,As)) ) ) ) ) ).

% one_assn_raw.pelims(1)
tff(fact_5409_Restr__natLeq,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bn,N)),aTP_Lamm_bo(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_bp(N))) ).

% Restr_natLeq
tff(fact_5410_Restr__natLeq2,axiom,
    ! [N: nat] : aa_set8868280134176360700at_nat(aa_set3608868651603078568at_nat(inf_in2572325071724192079at_nat,bNF_Ca8665028551170535155natLeq),produc457027306803732586at_nat(aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),N),aTP_Lamm_bq(N))) = collec3392354462482085612at_nat(produc6081775807080527818_nat_o(aTP_Lamm_bp(N))) ).

% Restr_natLeq2
tff(fact_5411_one__integer__code,axiom,
    one_one_Code_integer = code_Pos(one) ).

% one_integer_code
tff(fact_5412_one__natural_Orep__eq,axiom,
    code_nat_of_natural(one_one_Code_natural) = one_one_nat ).

% one_natural.rep_eq
tff(fact_5413_nat__of__num__code_I1_J,axiom,
    nat_of_num(one) = one_one_nat ).

% nat_of_num_code(1)
tff(fact_5414_one__int__code,axiom,
    one_one_int = pos(one) ).

% one_int_code
tff(fact_5415_one__natural_Otransfer,axiom,
    aa_Code_natural_o(code_pcr_natural(one_one_nat),one_one_Code_natural) ).

% one_natural.transfer
tff(fact_5416_one__integer_Otransfer,axiom,
    aa_Code_integer_o(code_pcr_integer(one_one_int),one_one_Code_integer) ).

% one_integer.transfer
tff(fact_5417_rat_Odomain__par,axiom,
    ! [DR1: fun_int_o,DR2: fun_int_o,P22: fun_Pr3238374608701176162_int_o] :
      ( ( domainp_int_int(fequal_int) = DR1 )
     => ( ( domainp_int_int(fequal_int) = DR2 )
       => ( bNF_re8699439704749558557nt_o_o(basic_4387203522000727145nt_int(fequal_int,fequal_int),fequal_o,P22,aTP_Lamm_br)
         => ( domain9213661015745956397nt_rat(pcr_rat) = aa_fun5349570537165229698_int_o(aa_fun4287570227111185249_int_o(inf_in1782598092406300404_int_o,basic_1567116559311922317nt_int(DR1,DR2)),P22) ) ) ) ) ).

% rat.domain_par
tff(fact_5418_ATP_Olambda__1,axiom,
    ! [Uu: product_prod_int_int] :
      aa_Pro6970504127699319548nt_int(aTP_Lamm_av,Uu) = $ite(aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu) = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(aa_Pro6799258434465214121nt_int(product_snd_int_int,Uu),aa_Pro6799258434465214121nt_int(product_fst_int_int,Uu))) ).

% ATP.lambda_1
tff(fact_5419_ATP_Olambda__2,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_am,Uu) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),aa_nat_nat(suc,zero_zero_nat)) ).

% ATP.lambda_2
tff(fact_5420_ATP_Olambda__3,axiom,
    ! [Uu: product_prod_int_int] :
      ( aa_Pro4592068669816069595_int_o(aTP_Lamm_br,Uu)
    <=> aa_Pro4592068669816069595_int_o(aa_Pro3952998155787061599_int_o(ratrel,Uu),Uu) ) ).

% ATP.lambda_3
tff(fact_5421_ATP_Olambda__4,axiom,
    ! [Uu: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_bl,Uu) = produc5538323210962509403atural(next,aTP_Lamm_bk(Uu)) ).

% ATP.lambda_4
tff(fact_5422_ATP_Olambda__5,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_at,Uu),Uua) = $ite(Uu = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_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_5
tff(fact_5423_ATP_Olambda__6,axiom,
    ! [Uu: int,Uua: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_aw,Uu),Uua) = $ite(Uua = zero_zero_int,product_Pair_int_int(zero_zero_int,one_one_int),product_Pair_int_int(Uu,Uua)) ).

% ATP.lambda_6
tff(fact_5424_ATP_Olambda__7,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_int(aa_Cod1242632294114111009er_int(aTP_Lamm_ay,Uu),Uua) = $let(
        l2: int,
        l2:= aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),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_7
tff(fact_5425_ATP_Olambda__8,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa_int_o(aTP_Lamm_bh(Uu),Uua)
    <=> ( aa_rat_o(ord_less_eq_rat(ring_1_of_int_rat(Uua)),Uu)
        & aa_rat_o(ord_less_rat(Uu),ring_1_of_int_rat(aa_int_int(aa_int_fun_int_int(plus_plus_int,Uua),one_one_int))) ) ) ).

% ATP.lambda_8
tff(fact_5426_ATP_Olambda__9,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ao,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_9
tff(fact_5427_ATP_Olambda__10,axiom,
    ! [Uu: assn,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_al(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uua) ) ) ).

% ATP.lambda_10
tff(fact_5428_ATP_Olambda__11,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ah(Uu),Uua)
    <=> ( aa_Pro653082453696054616_nat_o(in_range,Uua)
        & ~ aa_Pro653082453696054616_nat_o(Uu,Uua) ) ) ).

% ATP.lambda_11
tff(fact_5429_ATP_Olambda__12,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_num(aa_Cod6128727250572145423er_num(aTP_Lamm_ax,Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= plus_plus_num(l2,l2),
          $ite(Uua = zero_z3403309356797280102nteger,l3,plus_plus_num(l3,one)) ) ) ).

% ATP.lambda_12
tff(fact_5430_ATP_Olambda__13,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa_Code_integer_nat(aa_Cod5420483313623307717er_nat(aTP_Lamm_ba,Uu),Uua) = $let(
        l2: nat,
        l2:= 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_13
tff(fact_5431_ATP_Olambda__14,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ap,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_14
tff(fact_5432_ATP_Olambda__15,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bn,Uu),Uua)
    <=> aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu) ) ).

% ATP.lambda_15
tff(fact_5433_ATP_Olambda__16,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_nat(aa_nat_fun_nat_nat(aTP_Lamm_ak,Uu),Uua) = aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,Uua),Uu) ).

% ATP.lambda_16
tff(fact_5434_ATP_Olambda__17,axiom,
    ! [Uu: set_Pr1261947904930325089at_nat,Uua: product_prod_nat_nat] :
      ( aa_Pro6150637604445964947_nat_o(aTP_Lamm_ad(Uu),Uua)
    <=> member8440522571783428010at_nat(Uua,Uu) ) ).

% ATP.lambda_17
tff(fact_5435_ATP_Olambda__18,axiom,
    ! [Uu: set_list_nat,Uua: list_nat] :
      ( aa_list_nat_o(aTP_Lamm_ab(Uu),Uua)
    <=> member_list_nat(Uua,Uu) ) ).

% ATP.lambda_18
tff(fact_5436_ATP_Olambda__19,axiom,
    ! [Uu: set_nat,Uua: nat] :
      ( aa_nat_o(aTP_Lamm_ac(Uu),Uua)
    <=> member_nat2(Uua,Uu) ) ).

% ATP.lambda_19
tff(fact_5437_ATP_Olambda__20,axiom,
    ! [Uu: set_int,Uua: int] :
      ( aa_int_o(aTP_Lamm_a(Uu),Uua)
    <=> member_int2(Uua,Uu) ) ).

% ATP.lambda_20
tff(fact_5438_ATP_Olambda__21,axiom,
    ! [Uu: set_fu3555266933746557007_nat_o,Uua: fun_Pr6460944699146269935_nat_o] :
      ( aa_fun1675921005893230629at_o_o(aTP_Lamm_aa(Uu),Uua)
    <=> member7544390268018397336_nat_o(Uua,Uu) ) ).

% ATP.lambda_21
tff(fact_5439_ATP_Olambda__22,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_bq(Uu),Uua) = aa_nat_set_nat(order_underS_nat(bNF_Ca8665028551170535155natLeq),Uu) ).

% ATP.lambda_22
tff(fact_5440_ATP_Olambda__23,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_bk(Uu),Uua) = produc6639722614265839536atural(aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(plus_p4538020629002901425atural,Uua),aa_Cod4272041906116450446atural(aa_Cod5141017464274028979atural(times_2397367101498566445atural,Uu),numera5444537566228673987atural(bit1(bit0(bit0(bit1(bit0(bit1(bit0(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(bit1(one)))))))))))))))))))))))))))))))))) ).

% ATP.lambda_23
tff(fact_5441_ATP_Olambda__24,axiom,
    ! [Uu: code_natural,Uua: code_natural] : aa_Cod6352003912080677724atural(aTP_Lamm_bm(Uu),Uua) = produc6639722614265839536atural(modulo8411746178871703098atural(Uua,Uu)) ).

% ATP.lambda_24
tff(fact_5442_ATP_Olambda__25,axiom,
    ! [Uu: nat,Uua: nat] : aa_nat_set_nat(aTP_Lamm_bo(Uu),Uua) = collect_nat(aa_nat_fun_nat_o(aTP_Lamm_bn,Uu)) ).

% ATP.lambda_25
tff(fact_5443_ATP_Olambda__26,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_au(Uu),Uua),Uub) = $ite(aa_Code_integer_o(ord_le3102999989581377725nteger(numera6620942414471956472nteger(Uu)),Uub),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(plus_p5714425477246183910nteger,aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua)),one_one_Code_integer),minus_8373710615458151222nteger(Uub,numera6620942414471956472nteger(Uu))),produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(aa_Cod4161879620478718930nteger(times_3573771949741848930nteger,numera6620942414471956472nteger(bit0(one))),Uua),Uub)) ).

% ATP.lambda_26
tff(fact_5444_ATP_Olambda__27,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa_nat7603221163087113075at_nat(aa_nat8803788312118491388at_nat(aTP_Lamm_aq(Uu),Uua),Uub) = $ite(aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,numeral_numeral_nat(Uu)),Uub),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,numeral_numeral_nat(bit0(one))),Uua)),one_one_nat),aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uub),numeral_numeral_nat(Uu))),product_Pair_nat_nat(aa_nat_nat(aa_nat_fun_nat_nat(times_times_nat,numeral_numeral_nat(bit0(one))),Uua),Uub)) ).

% ATP.lambda_27
tff(fact_5445_ATP_Olambda__28,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa_int7462565939943104719nt_int(aa_int2565445551602696444nt_int(aTP_Lamm_ar(Uu),Uua),Uub) = $ite(aa_int_o(aa_int_fun_int_o(ord_less_eq_int,numeral_numeral_int(Uu)),Uub),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,numeral_numeral_int(bit0(one))),Uua)),one_one_int),aa_int_int(aa_int_fun_int_int(minus_minus_int,Uub),numeral_numeral_int(Uu))),product_Pair_int_int(aa_int_int(aa_int_fun_int_int(times_times_int,numeral_numeral_int(bit0(one))),Uua),Uub)) ).

% ATP.lambda_28
tff(fact_5446_ATP_Olambda__29,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bc(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uu),Uub))) ).

% ATP.lambda_29
tff(fact_5447_ATP_Olambda__30,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bd(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(abs_abs_Code_integer(Uu),Uub))) ).

% ATP.lambda_30
tff(fact_5448_ATP_Olambda__31,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod6977894654539884864nteger(aa_Cod8380929501330563836nteger(aTP_Lamm_bb(Uu),Uua),Uub) = $ite(Uub = zero_z3403309356797280102nteger,produc1086072967326762835nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),zero_z3403309356797280102nteger),produc1086072967326762835nteger(minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),one_one_Code_integer),minus_8373710615458151222nteger(Uu,Uub))) ).

% ATP.lambda_31
tff(fact_5449_ATP_Olambda__32,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa_Cod3168588076859124107eger_o(aa_Cod1096751537565085541eger_o(aTP_Lamm_az(Uu),Uua),Uub) = produc6677183202524767010eger_o($ite(aa_Code_integer_o(ord_le6747313008572928689nteger(zero_z3403309356797280102nteger),Uu),Uua,minus_8373710615458151222nteger(aa_Cod1783974652533396728nteger(uminus1351360451143612070nteger,Uua),Uub)),Uub = one_one_Code_integer) ).

% ATP.lambda_32
tff(fact_5450_ATP_Olambda__33,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bf(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = aa_nat_nat(aa_nat_fun_nat_nat(minus_minus_nat,Uu),one_one_nat) )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_33
tff(fact_5451_ATP_Olambda__34,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_bg(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( aa_nat_nat(aa_nat_fun_nat_nat(plus_plus_nat,groups4561878855575611511st_nat(Uub)),one_one_nat) = Uua ) ) ) ).

% ATP.lambda_34
tff(fact_5452_ATP_Olambda__35,axiom,
    ! [Uu: nat,Uua: nat,Uub: list_nat] :
      ( aa_list_nat_o(aa_nat5883842100386203267_nat_o(aTP_Lamm_be(Uu),Uua),Uub)
    <=> ( ( size_size_list_nat(Uub) = Uu )
        & ( groups4561878855575611511st_nat(Uub) = Uua ) ) ) ).

% ATP.lambda_35
tff(fact_5453_ATP_Olambda__36,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa_nat_o(aa_nat_fun_nat_o(aTP_Lamm_bp(Uu),Uua),Uub)
    <=> ( aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uua),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_nat,Uub),Uu)
        & aa_nat_o(aa_nat_fun_nat_o(ord_less_eq_nat,Uua),Uub) ) ) ).

% ATP.lambda_36
tff(fact_5454_ATP_Olambda__37,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_ai(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        | aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_37
tff(fact_5455_ATP_Olambda__38,axiom,
    ! [Uu: assn,Uua: assn,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(aTP_Lamm_aj(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uu),Uub)
        & aa_Pro653082453696054616_nat_o(aa_ass4243947775541419017_nat_o(rep_assn,Uua),Uub) ) ) ).

% ATP.lambda_38
tff(fact_5456_ATP_Olambda__39,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_af(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        | aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_39
tff(fact_5457_ATP_Olambda__40,axiom,
    ! [Uu: fun_Pr6460944699146269935_nat_o,Uua: fun_Pr6460944699146269935_nat_o,Uub: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aa_fun2638059832795814268_nat_o(aTP_Lamm_ae(Uu),Uua),Uub)
    <=> ( aa_Pro653082453696054616_nat_o(Uu,Uub)
        & aa_Pro653082453696054616_nat_o(Uua,Uub) ) ) ).

% ATP.lambda_40
tff(fact_5458_ATP_Olambda__41,axiom,
    ! [Uu: assn,Uua: product_unit] : aa_Product_unit_assn(aTP_Lamm_bj(Uu),Uua) = Uu ).

% ATP.lambda_41
tff(fact_5459_ATP_Olambda__42,axiom,
    ! [Uu: nat] : aa_nat_nat(aTP_Lamm_an,Uu) = Uu ).

% ATP.lambda_42
tff(fact_5460_ATP_Olambda__43,axiom,
    ! [Uu: int] : aa_int_int(aTP_Lamm_as,Uu) = Uu ).

% ATP.lambda_43
tff(fact_5461_ATP_Olambda__44,axiom,
    ! [Uu: product_unit] : aa_Product_unit_assn(aTP_Lamm_bi,Uu) = one_one_assn ).

% ATP.lambda_44
tff(fact_5462_ATP_Olambda__45,axiom,
    ! [Uu: produc3658429121746597890et_nat] :
      ( aa_Pro653082453696054616_nat_o(aTP_Lamm_ag,Uu)
    <=> $false ) ).

% ATP.lambda_45

% Helper facts (6)
tff(help_fNot_2_1_U,axiom,
    ! [P5: $o] :
      ( (P5)
      | aa_o_o(fNot,(P5)) ) ).

tff(help_fNot_1_1_U,axiom,
    ! [P5: $o] :
      ( ~ aa_o_o(fNot,(P5))
      | ~ (P5) ) ).

tff(help_fequal_2_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ( (X) != (Y) )
      | aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y)) ) ).

tff(help_fequal_1_1_fequal_001_Eo_T,axiom,
    ! [X: $o,Y: $o] :
      ( ~ aa_o_o(aa_o_fun_o_o(fequal_o,(X)),(Y))
      | ( (X) = (Y) ) ) ).

tff(help_fequal_2_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      | aa_int_o(aa_int_fun_int_o(fequal_int,X),Y) ) ).

tff(help_fequal_1_1_fequal_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ~ aa_int_o(aa_int_fun_int_o(fequal_int,X),Y)
      | ( X = Y ) ) ).

% Conjectures (1)
tff(conj_0,conjecture,
    aa_assn_o(fI_QUERY(p,q),f) ).

%------------------------------------------------------------------------------