TPTP Problem File: ITP218^1.p

%------------------------------------------------------------------------------
% File     : ITP218^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_List_Assn 00019_000444
% 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    : 0062_VEBT_List_Assn_00019_000444 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 4894 (2077 unt; 706 typ;   0 def)
%            Number of atoms       : 11867 (5909 equ;   2 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 38651 (1694   ~; 171   |; 957   &;30635   @)
%                                         (   0 <=>;5194  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   6 avg)
%            Number of types       :   54 (  53 usr)
%            Number of type conns  : 3002 (3002   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  657 ( 653 usr;  39 con; 0-6 aty)
%            Number of variables   : 12323 ( 588   ^;11173   !; 562   ?;12323   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-17 16:49:12.702
%------------------------------------------------------------------------------
% Could-be-implicit typings (53)
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(ty_n_t__Heap____Time____Monad__OHeap_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Assertions__Oassn_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__b_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__b_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__List__Olist_It__Assertions__Oassn_J,type,
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thf(ty_n_t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Ounit,type,
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thf(ty_n_t__List__Olist_Itf__b_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__b_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Typerep__Otyperep,type,
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thf(ty_n_t__Assertions__Oassn,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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thf(ty_n_tf__b,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (653)
thf(sy_c_Assertions_Oassn_OAbs__assn,type,
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thf(sy_c_Assertions_Oassn_ORep__assn,type,
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thf(sy_c_Assertions_Oentails,type,
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thf(sy_c_Assertions_Oentailst,type,
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thf(sy_c_Assertions_Oin__range,type,
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thf(sy_c_Assertions_Oin__range__rel,type,
    in_range_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Ois__pure__assn,type,
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thf(sy_c_Assertions_Oone__assn__raw,type,
    one_assn_raw: produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oone__assn__raw__rel,type,
    one_assn_raw_rel: produc3658429121746597890et_nat > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Oproper,type,
    proper: ( produc3658429121746597890et_nat > $o ) > $o ).

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

thf(sy_c_Assertions_Opure__assn__raw_001t__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_001t__Nat__Onat,type,
    pure_a825153325127701367it_nat: $o > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_OrelH,type,
    relH: set_nat > heap_e7401611519738050253t_unit > heap_e7401611519738050253t_unit > $o ).

thf(sy_c_Assertions_Otimes__assn__raw,type,
    times_assn_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Otimes__assn__raw__rel,type,
    times_assn_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

thf(sy_c_Assertions_Owand__assn,type,
    wand_assn: assn > assn > assn ).

thf(sy_c_Assertions_Owand__raw,type,
    wand_raw: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Assertions_Owand__raw__rel,type,
    wand_raw_rel: produc2732055786443039994et_nat > produc2732055786443039994et_nat > $o ).

thf(sy_c_Automation_OFI,type,
    fi: list_P8527749157015355191n_assn > assn > assn > assn > assn > assn > $o ).

thf(sy_c_Automation_OFI__QUERY,type,
    fI_QUERY: assn > assn > assn > $o ).

thf(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: list_P8527749157015355191n_assn > assn > assn > assn > $o ).

thf(sy_c_Automation_OSLN,type,
    sln: assn ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bNF_Gr8793815481675871596_nat_o: set_li630567559872716595_nat_o > ( produc3658429121746597890et_nat > $o ) > set_li630567559872716595_nat_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Int__Oint,type,
    bNF_Gr1870224194279859149ft_int: set_list_int > int > set_list_int ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Nat__Onat,type,
    bNF_Gr1872714664788909425ft_nat: set_list_nat > nat > set_list_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    bNF_Gr4113829767105464016n_assn: set_li5131720305576846103n_assn > produc6575502325842934193n_assn > set_li5131720305576846103n_assn ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
    bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__b,type,
    bNF_Greatest_Shift_b: set_list_b > b > set_list_b ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    bNF_Gr992270151760408296_nat_o: set_li630567559872716595_nat_o > list_P7985473006766602707_nat_o > set_Pr4532377907799695533_nat_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
    bNF_Gr6350390219475566417cc_int: set_list_int > list_int > set_int ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
    bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__b,type,
    bNF_Greatest_Succ_b: set_list_b > list_b > set_b ).

thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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thf(sy_c_Finite__Set_Ocard_001t__Int__Oint,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Assertions__Oassn,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
    finite_finite_int: set_int > $o ).

thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Fun_Ocomp_001_Eo_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Assertions__Oassn,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Assertions__Oassn_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Assertions__Oassn,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

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

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

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

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

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

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

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Assertions__Oassn,type,
    uminus_uminus_assn: assn > assn ).

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

thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
    zero_zero_int: int ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Heap_Oheap_Oarrays_001t__Product____Type__Ounit,type,
    arrays_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > list_nat ).

thf(sy_c_Heap_Oheap_Olim_001t__Product____Type__Ounit,type,
    lim_Product_unit: heap_e7401611519738050253t_unit > nat ).

thf(sy_c_Heap_Oheap_Orefs_001t__Product____Type__Ounit,type,
    refs_Product_unit: heap_e7401611519738050253t_unit > typerep > nat > nat ).

thf(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_T5738788834812785303t_unit ).

thf(sy_c_Hoare__Triple_Ohoare__triple_001t__Product____Type__Ounit,type,
    hoare_8945653483474564448t_unit: assn > heap_T5738788834812785303t_unit > ( product_unit > assn ) > $o ).

thf(sy_c_If_001t__Assertions__Oassn,type,
    if_assn: $o > assn > assn > assn ).

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

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

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

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

thf(sy_c_If_001t__Set__Oset_It__Nat__Onat_J,type,
    if_set_nat: $o > set_nat > set_nat > set_nat ).

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

thf(sy_c_Lattices_Oinf__class_Oinf_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    inf_in1318976480646536635_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001_Eo,type,
    inf_inf_o: $o > $o > $o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Assertions__Oassn,type,
    inf_inf_assn: assn > assn > assn ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
    inf_inf_int: int > int > int ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
    inf_inf_nat: nat > nat > nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Product____Type__Ounit,type,
    inf_inf_Product_unit: product_unit > product_unit > product_unit ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_Eo_J,type,
    inf_inf_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
    inf_inf_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_Eo,type,
    sup_sup_o: $o > $o > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Assertions__Oassn,type,
    sup_sup_assn: assn > assn > assn ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
    sup_sup_int: int > int > int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
    sup_sup_nat: nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Product____Type__Ounit,type,
    sup_sup_Product_unit: product_unit > product_unit > product_unit ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_Eo_J,type,
    sup_sup_set_o: set_o > set_o > set_o ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Assertions__Oassn_J,type,
    sup_sup_set_assn: set_assn > set_assn > set_assn ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001_Eo,type,
    lattic1921953407002678535_Max_o: set_o > $o ).

thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Int__Oint,type,
    lattic8263393255366662781ax_int: set_int > int ).

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

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001_Eo,type,
    lattic1973801136483472281_Min_o: set_o > $o ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Int__Oint,type,
    lattic8718645017227715691in_int: set_int > int ).

thf(sy_c_Lattices__Big_Olinorder__class_OMin_001t__Nat__Onat,type,
    lattic8721135487736765967in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Int__Oint,type,
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thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Nat__Onat,type,
    lattic4812295286663557444_o_nat: ( ( produc3658429121746597890et_nat > $o ) > nat ) > set_Pr4532377907799695533_nat_o > produc3658429121746597890et_nat > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Assertions__Oassn,type,
    lattic7708394068118247271o_assn: ( $o > assn ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Int__Oint,type,
    lattic2773365557947402859_o_int: ( $o > int ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001_Eo_001t__Nat__Onat,type,
    lattic2775856028456453135_o_nat: ( $o > nat ) > set_o > $o ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Assertions__Oassn,type,
    lattic2187264146484958483t_assn: ( nat > assn ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Int__Oint,type,
    lattic7444442490073309207at_int: ( nat > int ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Nat__Onat,type,
    lattic7446932960582359483at_nat: ( nat > nat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001_Eo,type,
    lattic4107685809792843317_fin_o: set_o > $o ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Assertions__Oassn,type,
    lattic47131356835913163n_assn: set_assn > assn ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Int__Oint,type,
    lattic5235898064620869839in_int: set_int > int ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Nat__Onat,type,
    lattic5238388535129920115in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Product____Type__Ounit,type,
    lattic1263872656861969706t_unit: set_Product_unit > product_unit ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3014633134055518761et_nat: set_set_nat > set_nat ).

thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lattic30941717366863870at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__order__set_001_Eo,type,
    lattic5087519243920114290_set_o: ( $o > $o > $o ) > ( $o > $o > $o ) > ( $o > $o > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Assertions__Oassn,type,
    lattic5623431474481994958t_assn: ( assn > assn > assn ) > ( assn > assn > $o ) > ( assn > assn > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Int__Oint,type,
    lattic6006661108824415698et_int: ( int > int > int ) > ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Nat__Onat,type,
    lattic6009151579333465974et_nat: ( nat > nat > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Product____Type__Ounit,type,
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thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3109210760196336428et_nat: ( set_nat > set_nat > set_nat ) > ( set_nat > set_nat > $o ) > ( set_nat > set_nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__order__set_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lattic6529551498584149819at_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__set_001_Eo,type,
    lattic7139874143898644262_set_o: ( $o > $o > $o ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Assertions__Oassn,type,
    lattic7983604933768736026t_assn: ( assn > assn > assn ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Nat__Onat,type,
    lattic1029310888574255042et_nat: ( nat > nat > nat ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Product____Type__Ounit,type,
    lattic356177490429882523t_unit: ( product_unit > product_unit > product_unit ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic6452893353811829624et_nat: ( set_nat > set_nat > set_nat ) > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__set_OF_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__set_OF_001_Eo,type,
    lattic3100085485418696562ce_F_o: ( $o > $o > $o ) > set_o > $o ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Assertions__Oassn,type,
    lattic7606391089408249806F_assn: ( assn > assn > assn ) > set_assn > assn ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Nat__Onat,type,
    lattic7742739596368939638_F_nat: ( nat > nat > nat ) > set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Product____Type__Ounit,type,
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thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__set_OF_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001_Eo,type,
    lattic1508158080041050831_fin_o: set_o > $o ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Assertions__Oassn,type,
    lattic2150320897289308081n_assn: set_assn > assn ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Int__Oint,type,
    lattic1091506334969745077in_int: set_int > int ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Nat__Onat,type,
    lattic1093996805478795353in_nat: set_nat > nat ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Product____Type__Ounit,type,
    lattic5294303975357428420t_unit: set_Product_unit > product_unit ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Nat__Onat_J,type,
    lattic3835124923745554447et_nat: set_set_nat > set_nat ).

thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    lattic1541023418247406232at_nat: set_se7855581050983116737at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Oall__interval__int,type,
    all_interval_int: ( int > $o ) > int > int > $o ).

thf(sy_c_List_Oall__interval__nat,type,
    all_interval_nat: ( nat > $o ) > nat > nat > $o ).

thf(sy_c_List_Oappend_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Assertions__Oassn,type,
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thf(sy_c_List_Obind_001tf__b_001tf__a,type,
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thf(sy_c_List_Obind_001tf__b_001tf__b,type,
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thf(sy_c_List_Obutlast_001tf__b,type,
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thf(sy_c_List_Oconcat_001t__Assertions__Oassn,type,
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thf(sy_c_List_Oconcat_001t__Int__Oint,type,
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thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
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thf(sy_c_List_Ocount__list_001tf__b,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
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thf(sy_c_List_Ofoldr_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_List_Olast_001tf__b,type,
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thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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    map_Pr7568062423562401049sn_int: ( produc6575502325842934193n_assn > int ) > list_P8527749157015355191n_assn > list_int ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
    map_Pr7570552894071451325sn_nat: ( produc6575502325842934193n_assn > nat ) > list_P8527749157015355191n_assn > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
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thf(sy_c_List_Olist_Omap_001tf__a_001t__Int__Oint,type,
    map_a_int: ( a > int ) > list_a > list_int ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Int__Oint_J,type,
    map_a_list_int: ( a > list_int ) > list_a > list_list_int ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Nat__Onat_J,type,
    map_a_list_nat: ( a > list_nat ) > list_a > list_list_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    map_a_7516433405034028541n_assn: ( a > list_P8527749157015355191n_assn ) > list_a > list_l6351802567095793725n_assn ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__a_J,type,
    map_a_list_a: ( a > list_a ) > list_a > list_list_a ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__b_J,type,
    map_a_list_b: ( a > list_b ) > list_a > list_list_b ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
    map_a_nat: ( a > nat ) > list_a > list_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
    map_a_a: ( a > a ) > list_a > list_a ).

thf(sy_c_List_Olist_Omap_001tf__a_001tf__b,type,
    map_a_b: ( a > b ) > list_a > list_b ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__Int__Oint,type,
    map_b_int: ( b > int ) > list_b > list_int ).

thf(sy_c_List_Olist_Omap_001tf__b_001t__Nat__Onat,type,
    map_b_nat: ( b > nat ) > list_b > list_nat ).

thf(sy_c_List_Olist_Omap_001tf__b_001tf__a,type,
    map_b_a: ( b > a ) > list_b > list_a ).

thf(sy_c_List_Olist_Omap_001tf__b_001tf__b,type,
    map_b_b: ( b > b ) > list_b > list_b ).

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    set_Pr1139785259514867910n_assn: list_P8527749157015355191n_assn > set_Pr5949110396991348497n_assn ).

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Olist__ex1_001t__Int__Oint,type,
    list_ex1_int: ( int > $o ) > list_int > $o ).

thf(sy_c_List_Olist__ex1_001t__Nat__Onat,type,
    list_ex1_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olist__ex1_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    list_e7761433933450087034n_assn: ( produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Olist__ex1_001tf__a,type,
    list_ex1_a: ( a > $o ) > list_a > $o ).

thf(sy_c_List_Olist__ex1_001tf__b,type,
    list_ex1_b: ( b > $o ) > list_b > $o ).

thf(sy_c_List_Olist__ex_001t__Int__Oint,type,
    list_ex_int: ( int > $o ) > list_int > $o ).

thf(sy_c_List_Olist__ex_001t__Nat__Onat,type,
    list_ex_nat: ( nat > $o ) > list_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__Int__Oint,type,
    listrelp_int_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__Nat__Onat,type,
    listrelp_int_nat: ( int > nat > $o ) > list_int > list_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Int__Oint_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    listre1516905941988846596n_assn: ( int > produc6575502325842934193n_assn > $o ) > list_int > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Int__Oint,type,
    listrelp_nat_int: ( nat > int > $o ) > list_nat > list_int > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Nat__Onat,type,
    listrelp_nat_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    listre8980241315090511200n_assn: ( nat > produc6575502325842934193n_assn > $o ) > list_nat > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001tf__a,type,
    listrelp_nat_a: ( nat > a > $o ) > list_nat > list_a > $o ).

thf(sy_c_List_Olistrelp_001t__Nat__Onat_001tf__b,type,
    listrelp_nat_b: ( nat > b > $o ) > list_nat > list_b > $o ).

thf(sy_c_List_Olistrelp_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Int__Oint,type,
    listre4656976334622418398sn_int: ( produc6575502325842934193n_assn > int > $o ) > list_P8527749157015355191n_assn > list_int > $o ).

thf(sy_c_List_Olistrelp_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
    listre4659466805131468674sn_nat: ( produc6575502325842934193n_assn > nat > $o ) > list_P8527749157015355191n_assn > list_nat > $o ).

thf(sy_c_List_Olistrelp_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    listre7738141641822031743n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Olistrelp_001tf__a_001t__Int__Oint,type,
    listrelp_a_int: ( a > int > $o ) > list_a > list_int > $o ).

thf(sy_c_List_Olistrelp_001tf__a_001t__Nat__Onat,type,
    listrelp_a_nat: ( a > nat > $o ) > list_a > list_nat > $o ).

thf(sy_c_List_Olistrelp_001tf__a_001tf__a,type,
    listrelp_a_a: ( a > a > $o ) > list_a > list_a > $o ).

thf(sy_c_List_Olistrelp_001tf__a_001tf__b,type,
    listrelp_a_b: ( a > b > $o ) > list_a > list_b > $o ).

thf(sy_c_List_Olistrelp_001tf__b_001t__Int__Oint,type,
    listrelp_b_int: ( b > int > $o ) > list_b > list_int > $o ).

thf(sy_c_List_Olistrelp_001tf__b_001t__Nat__Onat,type,
    listrelp_b_nat: ( b > nat > $o ) > list_b > list_nat > $o ).

thf(sy_c_List_Olistrelp_001tf__b_001tf__a,type,
    listrelp_b_a: ( b > a > $o ) > list_b > list_a > $o ).

thf(sy_c_List_Olistrelp_001tf__b_001tf__b,type,
    listrelp_b_b: ( b > b > $o ) > list_b > list_b > $o ).

thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
    map_tailrec_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Assertions__Oassn,type,
    map_ta2194206859901583947n_assn: ( produc6575502325842934193n_assn > assn ) > list_P8527749157015355191n_assn > list_assn ).

thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Int__Oint,type,
    map_ta8161051695879884879nt_int: ( int > int ) > list_int > list_int > list_int ).

thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Nat__Onat,type,
    map_ta8163542166388935155nt_nat: ( int > nat ) > list_int > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001t__Int__Oint_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    map_ta906617570451240910n_assn: ( int > produc6575502325842934193n_assn ) > list_int > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Int__Oint,type,
    map_ta7161697983978830323at_int: ( nat > int ) > list_nat > list_int > list_int ).

thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Nat__Onat,type,
    map_ta7164188454487880599at_nat: ( nat > nat ) > list_nat > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001t__Nat__Onat_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    map_ta8369952943552905514n_assn: ( nat > produc6575502325842934193n_assn ) > list_nat > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Omap__tailrec__rev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Assertions__Oassn,type,
    map_ta5611102776663852196n_assn: ( produc6575502325842934193n_assn > assn ) > list_P8527749157015355191n_assn > list_assn > list_assn ).

thf(sy_c_List_Omap__tailrec__rev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Int__Oint,type,
    map_ta4046687963084812712sn_int: ( produc6575502325842934193n_assn > int ) > list_P8527749157015355191n_assn > list_int > list_int ).

thf(sy_c_List_Omap__tailrec__rev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
    map_ta4049178433593862988sn_nat: ( produc6575502325842934193n_assn > nat ) > list_P8527749157015355191n_assn > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    map_ta6859916920478844725n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn ) > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__Int__Oint,type,
    map_ta8708341958415907829_a_int: ( a > int ) > list_a > list_int > list_int ).

thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__Nat__Onat,type,
    map_ta8710832428924958105_a_nat: ( a > nat ) > list_a > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001tf__a_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    map_ta4951362296667995304n_assn: ( a > produc6575502325842934193n_assn ) > list_a > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Omap__tailrec__rev_001tf__b_001t__Int__Oint,type,
    map_ta720414250517526518_b_int: ( b > int ) > list_b > list_int > list_int ).

thf(sy_c_List_Omap__tailrec__rev_001tf__b_001t__Nat__Onat,type,
    map_ta722904721026576794_b_nat: ( b > nat ) > list_b > list_nat > list_nat ).

thf(sy_c_List_Omap__tailrec__rev_001tf__b_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    map_ta1524144078346395495n_assn: ( b > produc6575502325842934193n_assn ) > list_b > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Omaps_001t__Int__Oint_001t__Int__Oint,type,
    maps_int_int: ( int > list_int ) > list_int > list_int ).

thf(sy_c_List_Omaps_001t__Int__Oint_001t__Nat__Onat,type,
    maps_int_nat: ( int > list_nat ) > list_int > list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Int__Oint,type,
    maps_nat_int: ( nat > list_int ) > list_nat > list_int ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001t__Nat__Onat,type,
    maps_nat_nat: ( nat > list_nat ) > list_nat > list_nat ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__a,type,
    maps_nat_a: ( nat > list_a ) > list_nat > list_a ).

thf(sy_c_List_Omaps_001t__Nat__Onat_001tf__b,type,
    maps_nat_b: ( nat > list_b ) > list_nat > list_b ).

thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Int__Oint,type,
    maps_P5983552678929597788sn_int: ( produc6575502325842934193n_assn > list_int ) > list_P8527749157015355191n_assn > list_int ).

thf(sy_c_List_Omaps_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
    maps_P5986043149438648064sn_nat: ( produc6575502325842934193n_assn > list_nat ) > list_P8527749157015355191n_assn > list_nat ).

thf(sy_c_List_Omaps_001tf__a_001t__Int__Oint,type,
    maps_a_int: ( a > list_int ) > list_a > list_int ).

thf(sy_c_List_Omaps_001tf__a_001t__Nat__Onat,type,
    maps_a_nat: ( a > list_nat ) > list_a > list_nat ).

thf(sy_c_List_Omaps_001tf__a_001tf__a,type,
    maps_a_a: ( a > list_a ) > list_a > list_a ).

thf(sy_c_List_Omaps_001tf__a_001tf__b,type,
    maps_a_b: ( a > list_b ) > list_a > list_b ).

thf(sy_c_List_Omaps_001tf__b_001t__Int__Oint,type,
    maps_b_int: ( b > list_int ) > list_b > list_int ).

thf(sy_c_List_Omaps_001tf__b_001t__Nat__Onat,type,
    maps_b_nat: ( b > list_nat ) > list_b > list_nat ).

thf(sy_c_List_Omaps_001tf__b_001tf__a,type,
    maps_b_a: ( b > list_a ) > list_b > list_a ).

thf(sy_c_List_Omaps_001tf__b_001tf__b,type,
    maps_b_b: ( b > list_b ) > list_b > list_b ).

thf(sy_c_List_Omember_001t__Int__Oint,type,
    member_int: list_int > int > $o ).

thf(sy_c_List_Omember_001t__Nat__Onat,type,
    member_nat: list_nat > nat > $o ).

thf(sy_c_List_Omember_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    member743271071679453132n_assn: list_P8527749157015355191n_assn > produc6575502325842934193n_assn > $o ).

thf(sy_c_List_Omember_001tf__a,type,
    member_a: list_a > a > $o ).

thf(sy_c_List_Omember_001tf__b,type,
    member_b: list_b > b > $o ).

thf(sy_c_List_On__lists_001t__Int__Oint,type,
    n_lists_int: nat > list_int > list_list_int ).

thf(sy_c_List_On__lists_001t__Nat__Onat,type,
    n_lists_nat: nat > list_nat > list_list_nat ).

thf(sy_c_List_On__lists_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    n_list679713369942834879n_assn: nat > list_P8527749157015355191n_assn > list_l6351802567095793725n_assn ).

thf(sy_c_List_On__lists_001tf__a,type,
    n_lists_a: nat > list_a > list_list_a ).

thf(sy_c_List_On__lists_001tf__b,type,
    n_lists_b: nat > list_b > list_list_b ).

thf(sy_c_List_Onth_001t__Assertions__Oassn,type,
    nth_assn: list_assn > nat > assn ).

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

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    nth_Pr1769885009046257848n_assn: list_P8527749157015355191n_assn > nat > produc6575502325842934193n_assn ).

thf(sy_c_List_Onth_001tf__a,type,
    nth_a: list_a > nat > a ).

thf(sy_c_List_Onth_001tf__b,type,
    nth_b: list_b > nat > b ).

thf(sy_c_List_Oord_Olexordp__eq_001t__Int__Oint,type,
    lexordp_eq_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(sy_c_List_Oord_Olexordp__eq_001t__Nat__Onat,type,
    lexordp_eq_nat: ( nat > nat > $o ) > list_nat > list_nat > $o ).

thf(sy_c_List_Oord_Olexordp__eq_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    lexord6224210647917505021n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Oord_Olexordp__eq_001tf__a,type,
    lexordp_eq_a: ( a > a > $o ) > list_a > list_a > $o ).

thf(sy_c_List_Oord_Olexordp__eq_001tf__b,type,
    lexordp_eq_b: ( b > b > $o ) > list_b > list_b > $o ).

thf(sy_c_List_Oord__class_Olexordp_001t__Assertions__Oassn,type,
    ord_lexordp_assn: list_assn > list_assn > $o ).

thf(sy_c_List_Oord__class_Olexordp_001t__Int__Oint,type,
    ord_lexordp_int: list_int > list_int > $o ).

thf(sy_c_List_Oord__class_Olexordp_001t__Nat__Onat,type,
    ord_lexordp_nat: list_nat > list_nat > $o ).

thf(sy_c_List_Oord__class_Olexordp__eq_001t__Assertions__Oassn,type,
    ord_lexordp_eq_assn: list_assn > list_assn > $o ).

thf(sy_c_List_Oord__class_Olexordp__eq_001t__Int__Oint,type,
    ord_lexordp_eq_int: list_int > list_int > $o ).

thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat,type,
    ord_lexordp_eq_nat: list_nat > list_nat > $o ).

thf(sy_c_List_Oproduct_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
    product_assn_assn: list_assn > list_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
    product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
    product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Int__Oint,type,
    product_nat_int: list_nat > list_int > list_P3521021558325789923at_int ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
    product_lists_int: list_list_int > list_list_int ).

thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    produc1566369527784499744n_assn: list_l6351802567095793725n_assn > list_l6351802567095793725n_assn ).

thf(sy_c_List_Oproduct__lists_001tf__a,type,
    product_lists_a: list_list_a > list_list_a ).

thf(sy_c_List_Oproduct__lists_001tf__b,type,
    product_lists_b: list_list_b > list_list_b ).

thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

thf(sy_c_List_Oremove1_001_Eo,type,
    remove1_o: $o > list_o > list_o ).

thf(sy_c_List_Oremove1_001t__Int__Oint,type,
    remove1_int: int > list_int > list_int ).

thf(sy_c_List_Oremove1_001t__Nat__Onat,type,
    remove1_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oremove1_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    remove1670527618125605709n_assn: produc6575502325842934193n_assn > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Oremove1_001tf__a,type,
    remove1_a: a > list_a > list_a ).

thf(sy_c_List_Oremove1_001tf__b,type,
    remove1_b: b > list_b > list_b ).

thf(sy_c_List_Orev_001t__Assertions__Oassn,type,
    rev_assn: list_assn > list_assn ).

thf(sy_c_List_Orev_001t__Int__Oint,type,
    rev_int: list_int > list_int ).

thf(sy_c_List_Orev_001t__List__Olist_It__Int__Oint_J,type,
    rev_list_int: list_list_int > list_list_int ).

thf(sy_c_List_Orev_001t__List__Olist_It__Nat__Onat_J,type,
    rev_list_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Orev_001t__Nat__Onat,type,
    rev_nat: list_nat > list_nat ).

thf(sy_c_List_Orev_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    rev_Pr4855572775806611735n_assn: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Orev_001tf__a,type,
    rev_a: list_a > list_a ).

thf(sy_c_List_Orev_001tf__b,type,
    rev_b: list_b > list_b ).

thf(sy_c_List_Orotate1_001t__Assertions__Oassn,type,
    rotate1_assn: list_assn > list_assn ).

thf(sy_c_List_Orotate1_001t__Int__Oint,type,
    rotate1_int: list_int > list_int ).

thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
    rotate1_nat: list_nat > list_nat ).

thf(sy_c_List_Orotate1_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    rotate328796349445179396n_assn: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_List_Orotate1_001tf__a,type,
    rotate1_a: list_a > list_a ).

thf(sy_c_List_Orotate1_001tf__b,type,
    rotate1_b: list_b > list_b ).

thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(sy_c_List_Osorted__wrt_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    sorted3986126766855596574n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > $o ).

thf(sy_c_List_Osorted__wrt_001tf__a,type,
    sorted_wrt_a: ( a > a > $o ) > list_a > $o ).

thf(sy_c_List_Osorted__wrt_001tf__b,type,
    sorted_wrt_b: ( b > b > $o ) > list_b > $o ).

thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    subseq184808802919281286n_assn: list_P8527749157015355191n_assn > list_l6351802567095793725n_assn ).

thf(sy_c_List_Osubseqs_001tf__a,type,
    subseqs_a: list_a > list_list_a ).

thf(sy_c_List_Osubseqs_001tf__b,type,
    subseqs_b: list_b > list_list_b ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

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

thf(sy_c_List_Oupto,type,
    upto: int > int > list_int ).

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

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

thf(sy_c_Misc_Olist__collect__set_001t__Assertions__Oassn_001_Eo,type,
    list_c312183563312650144assn_o: ( assn > set_o ) > list_assn > set_o ).

thf(sy_c_Misc_Olist__collect__set_001t__Assertions__Oassn_001t__Nat__Onat,type,
    list_c1844713377658005960sn_nat: ( assn > set_nat ) > list_assn > set_nat ).

thf(sy_c_Misc_Olist__collect__set_001t__Int__Oint_001_Eo,type,
    list_c6226808193739131804_int_o: ( int > set_o ) > list_int > set_o ).

thf(sy_c_Misc_Olist__collect__set_001t__Int__Oint_001t__Nat__Onat,type,
    list_c3451693981498911948nt_nat: ( int > set_nat ) > list_int > set_nat ).

thf(sy_c_Misc_Olist__collect__set_001t__Nat__Onat_001_Eo,type,
    list_c8047850539171819768_nat_o: ( nat > set_o ) > list_nat > set_o ).

thf(sy_c_Misc_Olist__collect__set_001t__Nat__Onat_001t__Nat__Onat,type,
    list_c2452340269597857392at_nat: ( nat > set_nat ) > list_nat > set_nat ).

thf(sy_c_Misc_Olist__collect__set_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001_Eo,type,
    list_c5102633440097552245assn_o: ( produc6575502325842934193n_assn > set_o ) > list_P8527749157015355191n_assn > set_o ).

thf(sy_c_Misc_Olist__collect__set_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_001t__Nat__Onat,type,
    list_c6061723043370948915sn_nat: ( produc6575502325842934193n_assn > set_nat ) > list_P8527749157015355191n_assn > set_nat ).

thf(sy_c_Misc_Olist__collect__set_001tf__a_001_Eo,type,
    list_collect_set_a_o: ( a > set_o ) > list_a > set_o ).

thf(sy_c_Misc_Olist__collect__set_001tf__a_001t__Nat__Onat,type,
    list_c5512459755930457856_a_nat: ( a > set_nat ) > list_a > set_nat ).

thf(sy_c_Misc_Olist__collect__set_001tf__b_001_Eo,type,
    list_collect_set_b_o: ( b > set_o ) > list_b > set_o ).

thf(sy_c_Misc_Olist__collect__set_001tf__b_001t__Nat__Onat,type,
    list_c6747904084886852353_b_nat: ( b > set_nat ) > list_b > set_nat ).

thf(sy_c_Misc_Omerge_001t__Int__Oint,type,
    merge_int: list_int > list_int > list_int ).

thf(sy_c_Misc_Omerge_001t__Nat__Onat,type,
    merge_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Misc_Omerge__list_001t__Int__Oint,type,
    merge_list_int: list_list_int > list_list_int > list_int ).

thf(sy_c_Misc_Omerge__list_001t__Nat__Onat,type,
    merge_list_nat: list_list_nat > list_list_nat > list_nat ).

thf(sy_c_Misc_Omergesort_001t__Assertions__Oassn,type,
    mergesort_assn: list_assn > list_assn ).

thf(sy_c_Misc_Omergesort_001t__Int__Oint,type,
    mergesort_int: list_int > list_int ).

thf(sy_c_Misc_Omergesort_001t__Nat__Onat,type,
    mergesort_nat: list_nat > list_nat ).

thf(sy_c_Misc_Omergesort_001t__Set__Oset_It__Nat__Onat_J,type,
    mergesort_set_nat: list_set_nat > list_set_nat ).

thf(sy_c_Misc_Omergesort__by__rel_001t__Assertions__Oassn,type,
    merges2400687674486622701l_assn: ( assn > assn > $o ) > list_assn > list_assn ).

thf(sy_c_Misc_Omergesort__by__rel_001t__Int__Oint,type,
    mergesort_by_rel_int: ( int > int > $o ) > list_int > list_int ).

thf(sy_c_Misc_Omergesort__by__rel_001t__Nat__Onat,type,
    mergesort_by_rel_nat: ( nat > nat > $o ) > list_nat > list_nat ).

thf(sy_c_Misc_Omergesort__by__rel_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    merges5609009838848816300n_assn: ( produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o ) > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_Misc_Omergesort__by__rel_001t__Set__Oset_It__Nat__Onat_J,type,
    merges7492048612793653835et_nat: ( set_nat > set_nat > $o ) > list_set_nat > list_set_nat ).

thf(sy_c_Misc_Omergesort__by__rel_001tf__a,type,
    mergesort_by_rel_a: ( a > a > $o ) > list_a > list_a ).

thf(sy_c_Misc_Omergesort__by__rel_001tf__b,type,
    mergesort_by_rel_b: ( b > b > $o ) > list_b > list_b ).

thf(sy_c_Misc_Orevg_001t__Int__Oint,type,
    revg_int: list_int > list_int > list_int ).

thf(sy_c_Misc_Orevg_001t__Nat__Onat,type,
    revg_nat: list_nat > list_nat > list_nat ).

thf(sy_c_Misc_Orevg_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    revg_P8856960164974728692n_assn: list_P8527749157015355191n_assn > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_Misc_Orevg_001tf__a,type,
    revg_a: list_a > list_a > list_a ).

thf(sy_c_Misc_Orevg_001tf__b,type,
    revg_b: list_b > list_b > list_b ).

thf(sy_c_Misc_Oslice_001t__Int__Oint,type,
    slice_int: nat > nat > list_int > list_int ).

thf(sy_c_Misc_Oslice_001t__Nat__Onat,type,
    slice_nat: nat > nat > list_nat > list_nat ).

thf(sy_c_Misc_Oslice_001t__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J,type,
    slice_7964966981652229398n_assn: nat > nat > list_P8527749157015355191n_assn > list_P8527749157015355191n_assn ).

thf(sy_c_Misc_Oslice_001tf__a,type,
    slice_a: nat > nat > list_a > list_a ).

thf(sy_c_Misc_Oslice_001tf__b,type,
    slice_b: nat > nat > list_b > list_b ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Assertions__Oassn_J,type,
    size_size_list_assn: list_assn > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Assertions__Oassn_Mt__Assertions__Oassn_J_J,type,
    size_s6829681357464350627n_assn: list_P8527749157015355191n_assn > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J_J,type,
    size_s7647898544948552527nt_nat: list_P8198026277950538467nt_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
    size_size_list_b: list_b > nat ).

thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
    neg_nu3811975205180677377ec_int: int > int ).

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

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

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

thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_M_Eo_J,type,
    bot_bo7963750851167320836at_o_o: ( produc3658429121746597890et_nat > $o ) > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_Eo_M_Eo_J,type,
    bot_bot_o_o: $o > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
    bot_bot_nat_o: nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Assertions__Oassn,type,
    bot_bot_assn: assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Product____Type__Ounit,type,
    bot_bot_Product_unit: product_unit ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    bot_bo7824918357723582233_nat_o: set_Pr4532377907799695533_nat_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Assertions__Oassn_J,type,
    bot_bot_set_assn: set_assn ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    bot_bo3957492148770167129t_unit: set_Product_unit ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    bot_bo3083307316010499117at_nat: set_se7855581050983116737at_nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_Eo,type,
    ord_less_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Assertions__Oassn,type,
    ord_less_assn: assn > assn > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Product____Type__Ounit,type,
    ord_le361264281704409273t_unit: product_unit > product_unit > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    ord_le2453136405763929_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_I_Eo_J,type,
    ord_less_set_o: set_o > set_o > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
    ord_le729326519192465773_nat_o: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
    ord_less_eq_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Assertions__Oassn,type,
    ord_less_eq_assn: assn > assn > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Product____Type__Ounit,type,
    ord_le3221252021190050221t_unit: product_unit > product_unit > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_J,type,
    ord_le2965882846123202637_nat_o: set_Pr4532377907799695533_nat_o > set_Pr4532377907799695533_nat_o > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Assertions__Oassn_J,type,
    ord_less_eq_set_assn: set_assn > set_assn > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    ord_le2077887516847798113at_nat: set_se7855581050983116737at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001_Eo,type,
    ord_max_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Assertions__Oassn,type,
    ord_max_assn: assn > assn > assn ).

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_I_Eo_J,type,
    ord_max_set_o: set_o > set_o > set_o ).

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_max_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Oord__class_Omin_001_Eo,type,
    ord_min_o: $o > $o > $o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Assertions__Oassn,type,
    ord_min_assn: assn > assn > assn ).

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

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

thf(sy_c_Orderings_Oord__class_Omin_001t__Product____Type__Ounit,type,
    ord_min_Product_unit: product_unit > product_unit > product_unit ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_I_Eo_J,type,
    ord_min_set_o: set_o > set_o > set_o ).

thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_min_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Assertions__Oassn,type,
    top_top_assn: assn ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Power_Opower__class_Opower_001t__Assertions__Oassn,type,
    power_power_assn: assn > nat > assn ).

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

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

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

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J_001t__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_OUnity,type,
    product_Unity: product_unit ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Assertions__Oassn_001t__Assertions__Oassn_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Set_OCollect_001_062_It__Product____Type__Oprod_It__Heap__Oheap__Oheap____ext_It__Product____Type__Ounit_J_Mt__Set__Oset_It__Nat__Onat_J_J_M_Eo_J,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Assertions__Oassn,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
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thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Ois__empty_001_Eo,type,
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thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
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thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
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thf(sy_c_Set_Oremove_001_Eo,type,
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thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
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thf(sy_c_Set_Othe__elem_001_Eo,type,
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thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
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thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
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thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
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thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Assertions__Oassn_001t__Assertions__Oassn,type,
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thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Syntax__Match_Osyntax__fo__nomatch_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001tf__a,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Int__Oint_001tf__b,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001tf__a,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001t__Nat__Onat_001tf__b,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__a_001t__Int__Oint,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__a_001t__Nat__Onat,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__a_001tf__a,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__a_001tf__b,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__b_001t__Int__Oint,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__b_001t__Nat__Onat,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__b_001tf__a,type,
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thf(sy_c_VEBT__List__Assn_Olist__assn_001tf__b_001tf__b,type,
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thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_member_001t__List__Olist_Itf__b_J,type,
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thf(sy_c_member_001t__Nat__Onat,type,
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thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001tf__a,type,
    member_a2: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b2: b > set_b > $o ).

thf(sy_v_P,type,
    p: a > b > assn ).

thf(sy_v_l_H,type,
    l: list_b ).

% Relevant facts (4174)
thf(fact_0_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( ( pure_assn @ P )
        = ( pure_assn @ Q ) )
      = ( P = Q ) ) ).

% pure_assn_eq_conv
thf(fact_1_list__assn_Osimps_I1_J,axiom,
    ! [P: a > a > assn] :
      ( ( vEBT_L4319891404334229443sn_a_a @ P @ nil_a @ nil_a )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_2_list__assn_Osimps_I1_J,axiom,
    ! [P: a > nat > assn] :
      ( ( vEBT_L2368420405249185739_a_nat @ P @ nil_a @ nil_nat )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_3_list__assn_Osimps_I1_J,axiom,
    ! [P: a > int > assn] :
      ( ( vEBT_L2365929934740135463_a_int @ P @ nil_a @ nil_int )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_4_list__assn_Osimps_I1_J,axiom,
    ! [P: b > a > assn] :
      ( ( vEBT_L1532435822361553410sn_b_a @ P @ nil_b @ nil_a )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_5_list__assn_Osimps_I1_J,axiom,
    ! [P: b > b > assn] :
      ( ( vEBT_L1532435822361553411sn_b_b @ P @ nil_b @ nil_b )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_6_list__assn_Osimps_I1_J,axiom,
    ! [P: b > nat > assn] :
      ( ( vEBT_L3603864734205580236_b_nat @ P @ nil_b @ nil_nat )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_7_list__assn_Osimps_I1_J,axiom,
    ! [P: b > int > assn] :
      ( ( vEBT_L3601374263696529960_b_int @ P @ nil_b @ nil_int )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_8_list__assn_Osimps_I1_J,axiom,
    ! [P: nat > a > assn] :
      ( ( vEBT_L6400351906427472169_nat_a @ P @ nil_nat @ nil_a )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_9_list__assn_Osimps_I1_J,axiom,
    ! [P: nat > b > assn] :
      ( ( vEBT_L6400351906427472170_nat_b @ P @ nil_nat @ nil_b )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_10_list__assn_Osimps_I1_J,axiom,
    ! [P: nat > nat > assn] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ nil_nat @ nil_nat )
      = one_one_assn ) ).

% list_assn.simps(1)
thf(fact_11_is__pure__assn__pure,axiom,
    ! [P: $o] : ( is_pure_assn @ ( pure_assn @ P ) ) ).

% is_pure_assn_pure
thf(fact_12_list__ex1__simps_I1_J,axiom,
    ! [P: a > $o] :
      ~ ( list_ex1_a @ P @ nil_a ) ).

% list_ex1_simps(1)
thf(fact_13_list__ex1__simps_I1_J,axiom,
    ! [P: b > $o] :
      ~ ( list_ex1_b @ P @ nil_b ) ).

% list_ex1_simps(1)
thf(fact_14_list__ex1__simps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > $o] :
      ~ ( list_e7761433933450087034n_assn @ P @ nil_Pr5671120429643327159n_assn ) ).

% list_ex1_simps(1)
thf(fact_15_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_16_list__ex1__simps_I1_J,axiom,
    ! [P: int > $o] :
      ~ ( list_ex1_int @ P @ nil_int ) ).

% list_ex1_simps(1)
thf(fact_17_bind__simps_I1_J,axiom,
    ! [F: a > list_a] :
      ( ( bind_a_a @ nil_a @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_18_bind__simps_I1_J,axiom,
    ! [F: a > list_b] :
      ( ( bind_a_b @ nil_a @ F )
      = nil_b ) ).

% bind_simps(1)
thf(fact_19_bind__simps_I1_J,axiom,
    ! [F: a > list_nat] :
      ( ( bind_a_nat @ nil_a @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_20_bind__simps_I1_J,axiom,
    ! [F: a > list_int] :
      ( ( bind_a_int @ nil_a @ F )
      = nil_int ) ).

% bind_simps(1)
thf(fact_21_bind__simps_I1_J,axiom,
    ! [F: b > list_a] :
      ( ( bind_b_a @ nil_b @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_22_bind__simps_I1_J,axiom,
    ! [F: b > list_b] :
      ( ( bind_b_b @ nil_b @ F )
      = nil_b ) ).

% bind_simps(1)
thf(fact_23_bind__simps_I1_J,axiom,
    ! [F: b > list_nat] :
      ( ( bind_b_nat @ nil_b @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_24_bind__simps_I1_J,axiom,
    ! [F: b > list_int] :
      ( ( bind_b_int @ nil_b @ F )
      = nil_int ) ).

% bind_simps(1)
thf(fact_25_bind__simps_I1_J,axiom,
    ! [F: nat > list_a] :
      ( ( bind_nat_a @ nil_nat @ F )
      = nil_a ) ).

% bind_simps(1)
thf(fact_26_bind__simps_I1_J,axiom,
    ! [F: nat > list_b] :
      ( ( bind_nat_b @ nil_nat @ F )
      = nil_b ) ).

% bind_simps(1)
thf(fact_27_member__rec_I2_J,axiom,
    ! [Y: a] :
      ~ ( member_a @ nil_a @ Y ) ).

% member_rec(2)
thf(fact_28_member__rec_I2_J,axiom,
    ! [Y: b] :
      ~ ( member_b @ nil_b @ Y ) ).

% member_rec(2)
thf(fact_29_member__rec_I2_J,axiom,
    ! [Y: produc6575502325842934193n_assn] :
      ~ ( member743271071679453132n_assn @ nil_Pr5671120429643327159n_assn @ Y ) ).

% member_rec(2)
thf(fact_30_member__rec_I2_J,axiom,
    ! [Y: nat] :
      ~ ( member_nat @ nil_nat @ Y ) ).

% member_rec(2)
thf(fact_31_member__rec_I2_J,axiom,
    ! [Y: int] :
      ~ ( member_int @ nil_int @ Y ) ).

% member_rec(2)
thf(fact_32_is__pure__assnE,axiom,
    ! [A: assn] :
      ( ( is_pure_assn @ A )
     => ~ ! [P2: $o] :
            ( A
           != ( pure_assn @ P2 ) ) ) ).

% is_pure_assnE
thf(fact_33_is__pure__assn__def,axiom,
    ( is_pure_assn
    = ( ^ [A2: assn] :
        ? [P3: $o] :
          ( A2
          = ( pure_assn @ P3 ) ) ) ) ).

% is_pure_assn_def
thf(fact_34_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_a @ N @ nil_a )
      = N ) ).

% gen_length_code(1)
thf(fact_35_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_b @ N @ nil_b )
      = N ) ).

% gen_length_code(1)
thf(fact_36_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le1182329135810803268n_assn @ N @ nil_Pr5671120429643327159n_assn )
      = N ) ).

% gen_length_code(1)
thf(fact_37_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_38_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_int @ N @ nil_int )
      = N ) ).

% gen_length_code(1)
thf(fact_39_list__assn_Osimps_I4_J,axiom,
    ! [Uu: b > nat > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L3603864734205580236_b_nat @ Uu @ nil_b @ ( cons_nat @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_40_list__assn_Osimps_I4_J,axiom,
    ! [Uu: nat > nat > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ Uu @ nil_nat @ ( cons_nat @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_41_list__assn_Osimps_I4_J,axiom,
    ! [Uu: int > nat > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ Uu @ nil_int @ ( cons_nat @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_42_list__assn_Osimps_I4_J,axiom,
    ! [Uu: b > int > assn,V: int,Va: list_int] :
      ( ( vEBT_L3601374263696529960_b_int @ Uu @ nil_b @ ( cons_int @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_43_list__assn_Osimps_I4_J,axiom,
    ! [Uu: nat > int > assn,V: int,Va: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ Uu @ nil_nat @ ( cons_int @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_44_list__assn_Osimps_I4_J,axiom,
    ! [Uu: int > int > assn,V: int,Va: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ Uu @ nil_int @ ( cons_int @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_45_list__assn_Osimps_I4_J,axiom,
    ! [Uu: a > b > assn,V: b,Va: list_b] :
      ( ( vEBT_L4319891404334229444sn_a_b @ Uu @ nil_a @ ( cons_b @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_46_list__assn_Osimps_I4_J,axiom,
    ! [Uu: b > a > assn,V: a,Va: list_a] :
      ( ( vEBT_L1532435822361553410sn_b_a @ Uu @ nil_b @ ( cons_a @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_47_list__assn_Osimps_I4_J,axiom,
    ! [Uu: a > nat > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L2368420405249185739_a_nat @ Uu @ nil_a @ ( cons_nat @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_48_list__assn_Osimps_I4_J,axiom,
    ! [Uu: a > int > assn,V: int,Va: list_int] :
      ( ( vEBT_L2365929934740135463_a_int @ Uu @ nil_a @ ( cons_int @ V @ Va ) )
      = bot_bot_assn ) ).

% list_assn.simps(4)
thf(fact_49_list__assn_Osimps_I3_J,axiom,
    ! [Uu: nat > a > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L6400351906427472169_nat_a @ Uu @ ( cons_nat @ V @ Va ) @ nil_a )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_50_list__assn_Osimps_I3_J,axiom,
    ! [Uu: nat > b > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L6400351906427472170_nat_b @ Uu @ ( cons_nat @ V @ Va ) @ nil_b )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_51_list__assn_Osimps_I3_J,axiom,
    ! [Uu: nat > nat > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ Uu @ ( cons_nat @ V @ Va ) @ nil_nat )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_52_list__assn_Osimps_I3_J,axiom,
    ! [Uu: nat > int > assn,V: nat,Va: list_nat] :
      ( ( vEBT_L8298612041380073281at_int @ Uu @ ( cons_nat @ V @ Va ) @ nil_int )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_53_list__assn_Osimps_I3_J,axiom,
    ! [Uu: int > a > assn,V: int,Va: list_int] :
      ( ( vEBT_L4155206938757026253_int_a @ Uu @ ( cons_int @ V @ Va ) @ nil_a )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_54_list__assn_Osimps_I3_J,axiom,
    ! [Uu: int > b > assn,V: int,Va: list_int] :
      ( ( vEBT_L4155206938757026254_int_b @ Uu @ ( cons_int @ V @ Va ) @ nil_b )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_55_list__assn_Osimps_I3_J,axiom,
    ! [Uu: int > nat > assn,V: int,Va: list_int] :
      ( ( vEBT_L77084186935402305nt_nat @ Uu @ ( cons_int @ V @ Va ) @ nil_nat )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_56_list__assn_Osimps_I3_J,axiom,
    ! [Uu: int > int > assn,V: int,Va: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ Uu @ ( cons_int @ V @ Va ) @ nil_int )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_57_list__assn_Osimps_I3_J,axiom,
    ! [Uu: a > b > assn,V: a,Va: list_a] :
      ( ( vEBT_L4319891404334229444sn_a_b @ Uu @ ( cons_a @ V @ Va ) @ nil_b )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_58_list__assn_Osimps_I3_J,axiom,
    ! [Uu: b > a > assn,V: b,Va: list_b] :
      ( ( vEBT_L1532435822361553410sn_b_a @ Uu @ ( cons_b @ V @ Va ) @ nil_a )
      = bot_bot_assn ) ).

% list_assn.simps(3)
thf(fact_59_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
thf(fact_60_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
thf(fact_61_list_Oinject,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn,Y21: produc6575502325842934193n_assn,Y22: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X21 @ X22 )
        = ( cons_P2971678138204555879n_assn @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_62_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = one_one_assn )
      = P ) ).

% pure_assn_eq_emp_iff
thf(fact_63_pure__true,axiom,
    ( ( pure_assn @ $true )
    = one_one_assn ) ).

% pure_true
thf(fact_64_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( ( pure_assn @ P )
        = bot_bot_assn )
      = ~ P ) ).

% pure_assn_eq_false_iff
thf(fact_65_pure__false,axiom,
    ( ( pure_assn @ $false )
    = bot_bot_assn ) ).

% pure_false
thf(fact_66_assn__basic__inequalities_I3_J,axiom,
    bot_bot_assn != one_one_assn ).

% assn_basic_inequalities(3)
thf(fact_67_member__rec_I1_J,axiom,
    ! [X: b,Xs: list_b,Y: b] :
      ( ( member_b @ ( cons_b @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_b @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_68_member__rec_I1_J,axiom,
    ! [X: a,Xs: list_a,Y: a] :
      ( ( member_a @ ( cons_a @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_a @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_69_member__rec_I1_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat] :
      ( ( member_nat @ ( cons_nat @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_nat @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_70_member__rec_I1_J,axiom,
    ! [X: int,Xs: list_int,Y: int] :
      ( ( member_int @ ( cons_int @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member_int @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_71_member__rec_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ( member743271071679453132n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Y )
      = ( ( X = Y )
        | ( member743271071679453132n_assn @ Xs @ Y ) ) ) ).

% member_rec(1)
thf(fact_72_is__pure__assn__basic__simps_I2_J,axiom,
    is_pure_assn @ one_one_assn ).

% is_pure_assn_basic_simps(2)
thf(fact_73_is__pure__assn__basic__simps_I1_J,axiom,
    is_pure_assn @ bot_bot_assn ).

% is_pure_assn_basic_simps(1)
thf(fact_74_transpose_Ocases,axiom,
    ! [X: list_list_a] :
      ( ( X != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X2: a,Xs2: list_a,Xss: list_list_a] :
              ( X
             != ( cons_list_a @ ( cons_a @ X2 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_75_transpose_Ocases,axiom,
    ! [X: list_list_b] :
      ( ( X != nil_list_b )
     => ( ! [Xss: list_list_b] :
            ( X
           != ( cons_list_b @ nil_b @ Xss ) )
       => ~ ! [X2: b,Xs2: list_b,Xss: list_list_b] :
              ( X
             != ( cons_list_b @ ( cons_b @ X2 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_76_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
thf(fact_77_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
thf(fact_78_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
thf(fact_79_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_80_not__Cons__self2,axiom,
    ! [X: int,Xs: list_int] :
      ( ( cons_int @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_81_not__Cons__self2,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( cons_P2971678138204555879n_assn @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_82_list__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
       => ( ! [X2: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_a @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_83_list__nonempty__induct,axiom,
    ! [Xs: list_b,P: list_b > $o] :
      ( ( Xs != nil_b )
     => ( ! [X2: b] : ( P @ ( cons_b @ X2 @ nil_b ) )
       => ( ! [X2: b,Xs2: list_b] :
              ( ( Xs2 != nil_b )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_b @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_84_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_nat @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_85_list__nonempty__induct,axiom,
    ! [Xs: list_int,P: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : ( P @ ( cons_int @ X2 @ nil_int ) )
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_int @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_86_list__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: list_P8527749157015355191n_assn > $o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X2 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( ( P @ Xs2 )
               => ( P @ ( cons_P2971678138204555879n_assn @ X2 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_87_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_88_list__induct2_H,axiom,
    ! [P: list_a > list_b > $o,Xs: list_a,Ys: list_b] :
      ( ( P @ nil_a @ nil_b )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_a @ ( cons_b @ Y2 @ Ys2 ) )
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_89_list__induct2_H,axiom,
    ! [P: list_b > list_a > $o,Xs: list_b,Ys: list_a] :
      ( ( P @ nil_b @ nil_a )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_b @ ( cons_a @ Y2 @ Ys2 ) )
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_90_list__induct2_H,axiom,
    ! [P: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
      ( ( P @ nil_b @ nil_b )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_b @ ( cons_b @ Y2 @ Ys2 ) )
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_91_list__induct2_H,axiom,
    ! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
      ( ( P @ nil_a @ nil_nat )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys2 ) )
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_92_list__induct2_H,axiom,
    ! [P: list_b > list_nat > $o,Xs: list_b,Ys: list_nat] :
      ( ( P @ nil_b @ nil_nat )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_b @ ( cons_nat @ Y2 @ Ys2 ) )
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_93_list__induct2_H,axiom,
    ! [P: list_a > list_int > $o,Xs: list_a,Ys: list_int] :
      ( ( P @ nil_a @ nil_int )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( cons_a @ X2 @ Xs2 ) @ nil_int )
       => ( ! [Y2: int,Ys2: list_int] : ( P @ nil_a @ ( cons_int @ Y2 @ Ys2 ) )
         => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_94_list__induct2_H,axiom,
    ! [P: list_b > list_int > $o,Xs: list_b,Ys: list_int] :
      ( ( P @ nil_b @ nil_int )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( cons_b @ X2 @ Xs2 ) @ nil_int )
       => ( ! [Y2: int,Ys2: list_int] : ( P @ nil_b @ ( cons_int @ Y2 @ Ys2 ) )
         => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_95_list__induct2_H,axiom,
    ! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
      ( ( P @ nil_nat @ nil_a )
     => ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys2 ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_96_list__induct2_H,axiom,
    ! [P: list_nat > list_b > $o,Xs: list_nat,Ys: list_b] :
      ( ( P @ nil_nat @ nil_b )
     => ( ! [X2: nat,Xs2: list_nat] : ( P @ ( cons_nat @ X2 @ Xs2 ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_nat @ ( cons_b @ Y2 @ Ys2 ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_97_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y3: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_98_neq__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
      = ( ? [Y3: b,Ys3: list_b] :
            ( Xs
            = ( cons_b @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_99_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y3: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_100_neq__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
      = ( ? [Y3: int,Ys3: list_int] :
            ( Xs
            = ( cons_int @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_101_neq__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
      = ( ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( Xs
            = ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_102_remdups__adj_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ( ! [X2: a] :
            ( X
           != ( cons_a @ X2 @ nil_a ) )
       => ~ ! [X2: a,Y2: a,Xs2: list_a] :
              ( X
             != ( cons_a @ X2 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_103_remdups__adj_Ocases,axiom,
    ! [X: list_b] :
      ( ( X != nil_b )
     => ( ! [X2: b] :
            ( X
           != ( cons_b @ X2 @ nil_b ) )
       => ~ ! [X2: b,Y2: b,Xs2: list_b] :
              ( X
             != ( cons_b @ X2 @ ( cons_b @ Y2 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_104_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
thf(fact_105_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
thf(fact_106_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
thf(fact_107_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X2: nat,Xs2: list_nat] :
          ( X
         != ( cons_nat @ X2 @ Xs2 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_108_min__list_Ocases,axiom,
    ! [X: list_int] :
      ( ! [X2: int,Xs2: list_int] :
          ( X
         != ( cons_int @ X2 @ Xs2 ) )
     => ( X = nil_int ) ) ).

% min_list.cases
thf(fact_109_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_110_list_Oexhaust,axiom,
    ! [Y: list_b] :
      ( ( Y != nil_b )
     => ~ ! [X212: b,X222: list_b] :
            ( Y
           != ( cons_b @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_111_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_112_list_Oexhaust,axiom,
    ! [Y: list_int] :
      ( ( Y != nil_int )
     => ~ ! [X212: int,X222: list_int] :
            ( Y
           != ( cons_int @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_113_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
thf(fact_114_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_115_list_OdiscI,axiom,
    ! [List: list_b,X21: b,X22: list_b] :
      ( ( List
        = ( cons_b @ X21 @ X22 ) )
     => ( List != nil_b ) ) ).

% list.discI
thf(fact_116_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_117_list_OdiscI,axiom,
    ! [List: list_int,X21: int,X22: list_int] :
      ( ( List
        = ( cons_int @ X21 @ X22 ) )
     => ( List != nil_int ) ) ).

% list.discI
thf(fact_118_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
thf(fact_119_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_120_list_Odistinct_I1_J,axiom,
    ! [X21: b,X22: list_b] :
      ( nil_b
     != ( cons_b @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_121_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_122_list_Odistinct_I1_J,axiom,
    ! [X21: int,X22: list_int] :
      ( nil_int
     != ( cons_int @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_123_list_Odistinct_I1_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( nil_Pr5671120429643327159n_assn
     != ( cons_P2971678138204555879n_assn @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_124_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_a > list_a > $o,R: a > a > $o,Xs: list_a,Ys: list_a] :
      ( ! [Xs2: list_a] : ( P @ Xs2 @ nil_a )
     => ( ! [X_1: list_a] : ( P @ nil_a @ X_1 )
       => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_a @ Y2 @ Ys2 ) )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_a @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_125_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_b > list_a > $o,R: b > a > $o,Xs: list_b,Ys: list_a] :
      ( ! [Xs2: list_b] : ( P @ Xs2 @ nil_a )
     => ( ! [X_1: list_a] : ( P @ nil_b @ X_1 )
       => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_a @ Y2 @ Ys2 ) )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_b @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_126_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_a > list_b > $o,R: a > b > $o,Xs: list_a,Ys: list_b] :
      ( ! [Xs2: list_a] : ( P @ Xs2 @ nil_b )
     => ( ! [X_1: list_b] : ( P @ nil_a @ X_1 )
       => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_b @ Y2 @ Ys2 ) )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_a @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_127_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_b > list_b > $o,R: b > b > $o,Xs: list_b,Ys: list_b] :
      ( ! [Xs2: list_b] : ( P @ Xs2 @ nil_b )
     => ( ! [X_1: list_b] : ( P @ nil_b @ X_1 )
       => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_b @ Y2 @ Ys2 ) )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_b @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_128_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_nat > list_a > $o,R: nat > a > $o,Xs: list_nat,Ys: list_a] :
      ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_a )
     => ( ! [X_1: list_a] : ( P @ nil_nat @ X_1 )
       => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_a @ Y2 @ Ys2 ) )
               => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_129_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_nat > list_b > $o,R: nat > b > $o,Xs: list_nat,Ys: list_b] :
      ( ! [Xs2: list_nat] : ( P @ Xs2 @ nil_b )
     => ( ! [X_1: list_b] : ( P @ nil_nat @ X_1 )
       => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_b @ Y2 @ Ys2 ) )
               => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_nat @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_130_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_int > list_a > $o,R: int > a > $o,Xs: list_int,Ys: list_a] :
      ( ! [Xs2: list_int] : ( P @ Xs2 @ nil_a )
     => ( ! [X_1: list_a] : ( P @ nil_int @ X_1 )
       => ( ! [X2: int,Xs2: list_int,Y2: a,Ys2: list_a] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_a @ Y2 @ Ys2 ) )
               => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: a,Ys2: list_a] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_int @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_131_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_int > list_b > $o,R: int > b > $o,Xs: list_int,Ys: list_b] :
      ( ! [Xs2: list_int] : ( P @ Xs2 @ nil_b )
     => ( ! [X_1: list_b] : ( P @ nil_int @ X_1 )
       => ( ! [X2: int,Xs2: list_int,Y2: b,Ys2: list_b] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_b @ Y2 @ Ys2 ) )
               => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: int,Xs2: list_int,Y2: b,Ys2: list_b] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_int @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_b @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_132_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_a > list_nat > $o,R: a > nat > $o,Xs: list_a,Ys: list_nat] :
      ( ! [Xs2: list_a] : ( P @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P @ nil_a @ X_1 )
       => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_nat @ Y2 @ Ys2 ) )
               => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_a @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_a @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_133_mergesort__by__rel__merge__induct,axiom,
    ! [P: list_b > list_nat > $o,R: b > nat > $o,Xs: list_b,Ys: list_nat] :
      ( ! [Xs2: list_b] : ( P @ Xs2 @ nil_nat )
     => ( ! [X_1: list_nat] : ( P @ nil_b @ X_1 )
       => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( P @ Xs2 @ ( cons_nat @ Y2 @ Ys2 ) )
               => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( ~ ( R @ X2 @ Y2 )
               => ( ( P @ ( cons_b @ X2 @ Xs2 ) @ Ys2 )
                 => ( P @ ( cons_b @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% mergesort_by_rel_merge_induct
thf(fact_134_mem__Collect__eq,axiom,
    ! [A: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ A @ ( collec939566748876313656_nat_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_135_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_136_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_137_Collect__mem__eq,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( collec939566748876313656_nat_o
        @ ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_138_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat2 @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_139_Collect__mem__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X3: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_140_Collect__cong,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec939566748876313656_nat_o @ P )
        = ( collec939566748876313656_nat_o @ Q ) ) ) ).

% Collect_cong
thf(fact_141_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_142_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P @ X2 )
          = ( Q @ X2 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_143_list__induct__first2,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
       => ( ! [X1: a,X23: a,Xs2: list_a] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_a @ X1 @ ( cons_a @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_144_list__induct__first2,axiom,
    ! [P: list_b > $o,Xs: list_b] :
      ( ( P @ nil_b )
     => ( ! [X2: b] : ( P @ ( cons_b @ X2 @ nil_b ) )
       => ( ! [X1: b,X23: b,Xs2: list_b] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_b @ X1 @ ( cons_b @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_145_list__induct__first2,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X1: nat,X23: nat,Xs2: list_nat] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_nat @ X1 @ ( cons_nat @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_146_list__induct__first2,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ( P @ nil_int )
     => ( ! [X2: int] : ( P @ ( cons_int @ X2 @ nil_int ) )
       => ( ! [X1: int,X23: int,Xs2: list_int] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_int @ X1 @ ( cons_int @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_147_list__induct__first2,axiom,
    ! [P: list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn] :
      ( ( P @ nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X2 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X1: produc6575502325842934193n_assn,X23: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( P @ Xs2 )
             => ( P @ ( cons_P2971678138204555879n_assn @ X1 @ ( cons_P2971678138204555879n_assn @ X23 @ Xs2 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_induct_first2
thf(fact_148_list__2pre__induct,axiom,
    ! [P: list_a > list_a > $o,W1: list_a,W2: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [E: a,W12: list_a,W22: list_a] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_a @ E @ W12 ) @ W22 ) )
       => ( ! [E: a,W13: list_a,W23: list_a] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_a @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_149_list__2pre__induct,axiom,
    ! [P: list_a > list_b > $o,W1: list_a,W2: list_b] :
      ( ( P @ nil_a @ nil_b )
     => ( ! [E: a,W12: list_a,W22: list_b] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_a @ E @ W12 ) @ W22 ) )
       => ( ! [E: b,W13: list_a,W23: list_b] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_b @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_150_list__2pre__induct,axiom,
    ! [P: list_b > list_a > $o,W1: list_b,W2: list_a] :
      ( ( P @ nil_b @ nil_a )
     => ( ! [E: b,W12: list_b,W22: list_a] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_b @ E @ W12 ) @ W22 ) )
       => ( ! [E: a,W13: list_b,W23: list_a] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_a @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_151_list__2pre__induct,axiom,
    ! [P: list_b > list_b > $o,W1: list_b,W2: list_b] :
      ( ( P @ nil_b @ nil_b )
     => ( ! [E: b,W12: list_b,W22: list_b] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_b @ E @ W12 ) @ W22 ) )
       => ( ! [E: b,W13: list_b,W23: list_b] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_b @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_152_list__2pre__induct,axiom,
    ! [P: list_a > list_nat > $o,W1: list_a,W2: list_nat] :
      ( ( P @ nil_a @ nil_nat )
     => ( ! [E: a,W12: list_a,W22: list_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_a @ E @ W12 ) @ W22 ) )
       => ( ! [E: nat,W13: list_a,W23: list_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_nat @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_153_list__2pre__induct,axiom,
    ! [P: list_b > list_nat > $o,W1: list_b,W2: list_nat] :
      ( ( P @ nil_b @ nil_nat )
     => ( ! [E: b,W12: list_b,W22: list_nat] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_b @ E @ W12 ) @ W22 ) )
       => ( ! [E: nat,W13: list_b,W23: list_nat] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_nat @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_154_list__2pre__induct,axiom,
    ! [P: list_a > list_int > $o,W1: list_a,W2: list_int] :
      ( ( P @ nil_a @ nil_int )
     => ( ! [E: a,W12: list_a,W22: list_int] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_a @ E @ W12 ) @ W22 ) )
       => ( ! [E: int,W13: list_a,W23: list_int] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_int @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_155_list__2pre__induct,axiom,
    ! [P: list_b > list_int > $o,W1: list_b,W2: list_int] :
      ( ( P @ nil_b @ nil_int )
     => ( ! [E: b,W12: list_b,W22: list_int] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_b @ E @ W12 ) @ W22 ) )
       => ( ! [E: int,W13: list_b,W23: list_int] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_int @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_156_list__2pre__induct,axiom,
    ! [P: list_nat > list_a > $o,W1: list_nat,W2: list_a] :
      ( ( P @ nil_nat @ nil_a )
     => ( ! [E: nat,W12: list_nat,W22: list_a] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: a,W13: list_nat,W23: list_a] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_a @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_157_list__2pre__induct,axiom,
    ! [P: list_nat > list_b > $o,W1: list_nat,W2: list_b] :
      ( ( P @ nil_nat @ nil_b )
     => ( ! [E: nat,W12: list_nat,W22: list_b] :
            ( ( P @ W12 @ W22 )
           => ( P @ ( cons_nat @ E @ W12 ) @ W22 ) )
       => ( ! [E: b,W13: list_nat,W23: list_b] :
              ( ( P @ W13 @ W23 )
             => ( P @ W13 @ ( cons_b @ E @ W23 ) ) )
         => ( P @ W1 @ W2 ) ) ) ) ).

% list_2pre_induct
thf(fact_158_neq__NilE,axiom,
    ! [L: list_a] :
      ( ( L != nil_a )
     => ~ ! [X2: a,Xs2: list_a] :
            ( L
           != ( cons_a @ X2 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_159_neq__NilE,axiom,
    ! [L: list_b] :
      ( ( L != nil_b )
     => ~ ! [X2: b,Xs2: list_b] :
            ( L
           != ( cons_b @ X2 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_160_neq__NilE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] :
            ( L
           != ( cons_nat @ X2 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_161_neq__NilE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [X2: int,Xs2: list_int] :
            ( L
           != ( cons_int @ X2 @ Xs2 ) ) ) ).

% neq_NilE
thf(fact_162_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
thf(fact_163_quicksort_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X2: nat,Xs2: list_nat] :
            ( X
           != ( cons_nat @ X2 @ Xs2 ) ) ) ).

% quicksort.cases
thf(fact_164_quicksort_Ocases,axiom,
    ! [X: list_int] :
      ( ( X != nil_int )
     => ~ ! [X2: int,Xs2: list_int] :
            ( X
           != ( cons_int @ X2 @ Xs2 ) ) ) ).

% quicksort.cases
thf(fact_165_list__assn_Oelims,axiom,
    ! [X: b > b > assn,Xa: list_b,Xb: list_b,Y: assn] :
      ( ( ( vEBT_L1532435822361553411sn_b_b @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb
                    = ( cons_b @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L1532435822361553411sn_b_b @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: b,Va2: list_b] :
                  ( Xa
                  = ( cons_b @ V2 @ Va2 ) )
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: b,Va2: list_b] :
                      ( Xb
                      = ( cons_b @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_166_list__assn_Oelims,axiom,
    ! [X: b > nat > assn,Xa: list_b,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L3603864734205580236_b_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L3603864734205580236_b_nat @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: b,Va2: list_b] :
                  ( Xa
                  = ( cons_b @ V2 @ Va2 ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: nat,Va2: list_nat] :
                      ( Xb
                      = ( cons_nat @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_167_list__assn_Oelims,axiom,
    ! [X: b > int > assn,Xa: list_b,Xb: list_int,Y: assn] :
      ( ( ( vEBT_L3601374263696529960_b_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ! [C: int,Cs: list_int] :
                  ( ( Xb
                    = ( cons_int @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L3601374263696529960_b_int @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: b,Va2: list_b] :
                  ( Xa
                  = ( cons_b @ V2 @ Va2 ) )
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_b )
               => ( ? [V2: int,Va2: list_int] :
                      ( Xb
                      = ( cons_int @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_168_list__assn_Oelims,axiom,
    ! [X: nat > a > assn,Xa: list_nat,Xb: list_a,Y: assn] :
      ( ( ( vEBT_L6400351906427472169_nat_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_a )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C: a,Cs: list_a] :
                  ( ( Xb
                    = ( cons_a @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L6400351906427472169_nat_a @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] :
                  ( Xa
                  = ( cons_nat @ V2 @ Va2 ) )
             => ( ( Xb = nil_a )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: a,Va2: list_a] :
                      ( Xb
                      = ( cons_a @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_169_list__assn_Oelims,axiom,
    ! [X: nat > b > assn,Xa: list_nat,Xb: list_b,Y: assn] :
      ( ( ( vEBT_L6400351906427472170_nat_b @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb
                    = ( cons_b @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L6400351906427472170_nat_b @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] :
                  ( Xa
                  = ( cons_nat @ V2 @ Va2 ) )
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: b,Va2: list_b] :
                      ( Xb
                      = ( cons_b @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_170_list__assn_Oelims,axiom,
    ! [X: nat > nat > assn,Xa: list_nat,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L8301102511889123557at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L8301102511889123557at_nat @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] :
                  ( Xa
                  = ( cons_nat @ V2 @ Va2 ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: nat,Va2: list_nat] :
                      ( Xb
                      = ( cons_nat @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_171_list__assn_Oelims,axiom,
    ! [X: nat > int > assn,Xa: list_nat,Xb: list_int,Y: assn] :
      ( ( ( vEBT_L8298612041380073281at_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( ( Xb = nil_int )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ! [C: int,Cs: list_int] :
                  ( ( Xb
                    = ( cons_int @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L8298612041380073281at_int @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: nat,Va2: list_nat] :
                  ( Xa
                  = ( cons_nat @ V2 @ Va2 ) )
             => ( ( Xb = nil_int )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_nat )
               => ( ? [V2: int,Va2: list_int] :
                      ( Xb
                      = ( cons_int @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_172_list__assn_Oelims,axiom,
    ! [X: int > a > assn,Xa: list_int,Xb: list_a,Y: assn] :
      ( ( ( vEBT_L4155206938757026253_int_a @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_a )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C: a,Cs: list_a] :
                  ( ( Xb
                    = ( cons_a @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L4155206938757026253_int_a @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: int,Va2: list_int] :
                  ( Xa
                  = ( cons_int @ V2 @ Va2 ) )
             => ( ( Xb = nil_a )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: a,Va2: list_a] :
                      ( Xb
                      = ( cons_a @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_173_list__assn_Oelims,axiom,
    ! [X: int > b > assn,Xa: list_int,Xb: list_b,Y: assn] :
      ( ( ( vEBT_L4155206938757026254_int_b @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_b )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C: b,Cs: list_b] :
                  ( ( Xb
                    = ( cons_b @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L4155206938757026254_int_b @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: int,Va2: list_int] :
                  ( Xa
                  = ( cons_int @ V2 @ Va2 ) )
             => ( ( Xb = nil_b )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: b,Va2: list_b] :
                      ( Xb
                      = ( cons_b @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_174_list__assn_Oelims,axiom,
    ! [X: int > nat > assn,Xa: list_int,Xb: list_nat,Y: assn] :
      ( ( ( vEBT_L77084186935402305nt_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( ( Xb = nil_nat )
           => ( Y != one_one_assn ) ) )
       => ( ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ! [C: nat,Cs: list_nat] :
                  ( ( Xb
                    = ( cons_nat @ C @ Cs ) )
                 => ( Y
                   != ( times_times_assn @ ( X @ A4 @ C ) @ ( vEBT_L77084186935402305nt_nat @ X @ As @ Cs ) ) ) ) )
         => ( ( ? [V2: int,Va2: list_int] :
                  ( Xa
                  = ( cons_int @ V2 @ Va2 ) )
             => ( ( Xb = nil_nat )
               => ( Y != bot_bot_assn ) ) )
           => ~ ( ( Xa = nil_int )
               => ( ? [V2: nat,Va2: list_nat] :
                      ( Xb
                      = ( cons_nat @ V2 @ Va2 ) )
                 => ( Y != bot_bot_assn ) ) ) ) ) ) ) ).

% list_assn.elims
thf(fact_175_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_176_mult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% mult_1
thf(fact_177_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_178_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_179_mult_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.right_neutral
thf(fact_180_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_181_merge__pure__star,axiom,
    ! [A: $o,B: $o] :
      ( ( times_times_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_star
thf(fact_182_star__false__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ bot_bot_assn )
      = bot_bot_assn ) ).

% star_false_right
thf(fact_183_star__false__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ bot_bot_assn @ P )
      = bot_bot_assn ) ).

% star_false_left
thf(fact_184_is__pure__assn__starI,axiom,
    ! [A: assn,B: assn] :
      ( ( is_pure_assn @ A )
     => ( ( is_pure_assn @ B )
       => ( is_pure_assn @ ( times_times_assn @ A @ B ) ) ) ) ).

% is_pure_assn_starI
thf(fact_185_assn__times__comm,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q2: assn] : ( times_times_assn @ Q2 @ P3 ) ) ) ).

% assn_times_comm
thf(fact_186_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( times_times_assn @ ( times_times_assn @ P @ Q ) @ R )
      = ( times_times_assn @ P @ ( times_times_assn @ Q @ R ) ) ) ).

% assn_times_assoc
thf(fact_187_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( times_times_assn @ B @ ( times_times_assn @ A @ C2 ) )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_188_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult.left_commute
thf(fact_189_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_190_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_mult_class.mult.commute
thf(fact_191_mult_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_192_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_193_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_194_mult_Ocomm__neutral,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% mult.comm_neutral
thf(fact_195_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_196_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_197_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_198_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_199_assn__one__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ one_one_assn @ P )
      = P ) ).

% assn_one_left
thf(fact_200_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > nat > assn,A: nat,As2: list_nat,C2: nat,Cs2: list_nat] :
      ( ( vEBT_L8301102511889123557at_nat @ P @ ( cons_nat @ A @ As2 ) @ ( cons_nat @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L8301102511889123557at_nat @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_201_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > int > assn,A: nat,As2: list_nat,C2: int,Cs2: list_int] :
      ( ( vEBT_L8298612041380073281at_int @ P @ ( cons_nat @ A @ As2 ) @ ( cons_int @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L8298612041380073281at_int @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_202_list__assn_Osimps_I2_J,axiom,
    ! [P: int > nat > assn,A: int,As2: list_int,C2: nat,Cs2: list_nat] :
      ( ( vEBT_L77084186935402305nt_nat @ P @ ( cons_int @ A @ As2 ) @ ( cons_nat @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L77084186935402305nt_nat @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_203_list__assn_Osimps_I2_J,axiom,
    ! [P: int > int > assn,A: int,As2: list_int,C2: int,Cs2: list_int] :
      ( ( vEBT_L74593716426352029nt_int @ P @ ( cons_int @ A @ As2 ) @ ( cons_int @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L74593716426352029nt_int @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_204_list__assn_Osimps_I2_J,axiom,
    ! [P: a > b > assn,A: a,As2: list_a,C2: b,Cs2: list_b] :
      ( ( vEBT_L4319891404334229444sn_a_b @ P @ ( cons_a @ A @ As2 ) @ ( cons_b @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L4319891404334229444sn_a_b @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_205_list__assn_Osimps_I2_J,axiom,
    ! [P: b > a > assn,A: b,As2: list_b,C2: a,Cs2: list_a] :
      ( ( vEBT_L1532435822361553410sn_b_a @ P @ ( cons_b @ A @ As2 ) @ ( cons_a @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L1532435822361553410sn_b_a @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_206_list__assn_Osimps_I2_J,axiom,
    ! [P: a > nat > assn,A: a,As2: list_a,C2: nat,Cs2: list_nat] :
      ( ( vEBT_L2368420405249185739_a_nat @ P @ ( cons_a @ A @ As2 ) @ ( cons_nat @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L2368420405249185739_a_nat @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_207_list__assn_Osimps_I2_J,axiom,
    ! [P: a > int > assn,A: a,As2: list_a,C2: int,Cs2: list_int] :
      ( ( vEBT_L2365929934740135463_a_int @ P @ ( cons_a @ A @ As2 ) @ ( cons_int @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L2365929934740135463_a_int @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_208_list__assn_Osimps_I2_J,axiom,
    ! [P: a > a > assn,A: a,As2: list_a,C2: a,Cs2: list_a] :
      ( ( vEBT_L4319891404334229443sn_a_a @ P @ ( cons_a @ A @ As2 ) @ ( cons_a @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L4319891404334229443sn_a_a @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_209_list__assn_Osimps_I2_J,axiom,
    ! [P: nat > produc6575502325842934193n_assn > assn,A: nat,As2: list_nat,C2: produc6575502325842934193n_assn,Cs2: list_P8527749157015355191n_assn] :
      ( ( vEBT_L9083610150846024284n_assn @ P @ ( cons_nat @ A @ As2 ) @ ( cons_P2971678138204555879n_assn @ C2 @ Cs2 ) )
      = ( times_times_assn @ ( P @ A @ C2 ) @ ( vEBT_L9083610150846024284n_assn @ P @ As2 @ Cs2 ) ) ) ).

% list_assn.simps(2)
thf(fact_210_one__reorient,axiom,
    ! [X: assn] :
      ( ( one_one_assn = X )
      = ( X = one_one_assn ) ) ).

% one_reorient
thf(fact_211_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_212_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_213_list__tail__coinc,axiom,
    ! [N1: nat,R1: list_nat,N2: nat,R2: list_nat] :
      ( ( ( cons_nat @ N1 @ R1 )
        = ( cons_nat @ N2 @ R2 ) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
thf(fact_214_list__tail__coinc,axiom,
    ! [N1: int,R1: list_int,N2: int,R2: list_int] :
      ( ( ( cons_int @ N1 @ R1 )
        = ( cons_int @ N2 @ R2 ) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
thf(fact_215_list__tail__coinc,axiom,
    ! [N1: produc6575502325842934193n_assn,R1: list_P8527749157015355191n_assn,N2: produc6575502325842934193n_assn,R2: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ N1 @ R1 )
        = ( cons_P2971678138204555879n_assn @ N2 @ R2 ) )
     => ( ( N1 = N2 )
        & ( R1 = R2 ) ) ) ).

% list_tail_coinc
thf(fact_216_norm__assertion__simps_I1_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ one_one_assn @ A )
      = A ) ).

% norm_assertion_simps(1)
thf(fact_217_norm__assertion__simps_I2_J,axiom,
    ! [A: assn] :
      ( ( times_times_assn @ A @ one_one_assn )
      = A ) ).

% norm_assertion_simps(2)
thf(fact_218_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_a @ nil_list_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% product_lists.simps(1)
thf(fact_219_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_b @ nil_list_b )
    = ( cons_list_b @ nil_b @ nil_list_b ) ) ).

% product_lists.simps(1)
thf(fact_220_product__lists_Osimps_I1_J,axiom,
    ( ( produc1566369527784499744n_assn @ nil_li5476096274760905021n_assn )
    = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) ).

% product_lists.simps(1)
thf(fact_221_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_222_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_int @ nil_list_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% product_lists.simps(1)
thf(fact_223_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_a @ nil_a )
    = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% subseqs.simps(1)
thf(fact_224_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_b @ nil_b )
    = ( cons_list_b @ nil_b @ nil_list_b ) ) ).

% subseqs.simps(1)
thf(fact_225_subseqs_Osimps_I1_J,axiom,
    ( ( subseq184808802919281286n_assn @ nil_Pr5671120429643327159n_assn )
    = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) ).

% subseqs.simps(1)
thf(fact_226_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_227_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_int @ nil_int )
    = ( cons_list_int @ nil_int @ nil_list_int ) ) ).

% subseqs.simps(1)
thf(fact_228_insert__Nil,axiom,
    ! [X: a] :
      ( ( insert_a @ X @ nil_a )
      = ( cons_a @ X @ nil_a ) ) ).

% insert_Nil
thf(fact_229_insert__Nil,axiom,
    ! [X: b] :
      ( ( insert_b @ X @ nil_b )
      = ( cons_b @ X @ nil_b ) ) ).

% insert_Nil
thf(fact_230_insert__Nil,axiom,
    ! [X: nat] :
      ( ( insert_nat @ X @ nil_nat )
      = ( cons_nat @ X @ nil_nat ) ) ).

% insert_Nil
thf(fact_231_insert__Nil,axiom,
    ! [X: int] :
      ( ( insert_int @ X @ nil_int )
      = ( cons_int @ X @ nil_int ) ) ).

% insert_Nil
thf(fact_232_insert__Nil,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( insert3246601298802261197n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ).

% insert_Nil
thf(fact_233_revg_Oelims,axiom,
    ! [X: list_a,Xa: list_a,Y: list_a] :
      ( ( ( revg_a @ X @ Xa )
        = Y )
     => ( ( ( X = nil_a )
         => ( Y != Xa ) )
       => ~ ! [A4: a,As: list_a] :
              ( ( X
                = ( cons_a @ A4 @ As ) )
             => ( Y
               != ( revg_a @ As @ ( cons_a @ A4 @ Xa ) ) ) ) ) ) ).

% revg.elims
thf(fact_234_revg_Oelims,axiom,
    ! [X: list_b,Xa: list_b,Y: list_b] :
      ( ( ( revg_b @ X @ Xa )
        = Y )
     => ( ( ( X = nil_b )
         => ( Y != Xa ) )
       => ~ ! [A4: b,As: list_b] :
              ( ( X
                = ( cons_b @ A4 @ As ) )
             => ( Y
               != ( revg_b @ As @ ( cons_b @ A4 @ Xa ) ) ) ) ) ) ).

% revg.elims
thf(fact_235_revg_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( ( revg_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ~ ! [A4: nat,As: list_nat] :
              ( ( X
                = ( cons_nat @ A4 @ As ) )
             => ( Y
               != ( revg_nat @ As @ ( cons_nat @ A4 @ Xa ) ) ) ) ) ) ).

% revg.elims
thf(fact_236_revg_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( ( revg_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ~ ! [A4: int,As: list_int] :
              ( ( X
                = ( cons_int @ A4 @ As ) )
             => ( Y
               != ( revg_int @ As @ ( cons_int @ A4 @ Xa ) ) ) ) ) ) ).

% revg.elims
thf(fact_237_revg_Oelims,axiom,
    ! [X: list_P8527749157015355191n_assn,Xa: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( ( revg_P8856960164974728692n_assn @ X @ Xa )
        = Y )
     => ( ( ( X = nil_Pr5671120429643327159n_assn )
         => ( Y != Xa ) )
       => ~ ! [A4: produc6575502325842934193n_assn,As: list_P8527749157015355191n_assn] :
              ( ( X
                = ( cons_P2971678138204555879n_assn @ A4 @ As ) )
             => ( Y
               != ( revg_P8856960164974728692n_assn @ As @ ( cons_P2971678138204555879n_assn @ A4 @ Xa ) ) ) ) ) ) ).

% revg.elims
thf(fact_238_map__tailrec__rev_Oelims,axiom,
    ! [X: a > nat,Xa: list_a,Xb: list_nat,Y: list_nat] :
      ( ( ( map_ta8710832428924958105_a_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A4: a,As: list_a] :
              ( ( Xa
                = ( cons_a @ A4 @ As ) )
             => ( Y
               != ( map_ta8710832428924958105_a_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_239_map__tailrec__rev_Oelims,axiom,
    ! [X: b > nat,Xa: list_b,Xb: list_nat,Y: list_nat] :
      ( ( ( map_ta722904721026576794_b_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ( Y
               != ( map_ta722904721026576794_b_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_240_map__tailrec__rev_Oelims,axiom,
    ! [X: a > int,Xa: list_a,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8708341958415907829_a_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A4: a,As: list_a] :
              ( ( Xa
                = ( cons_a @ A4 @ As ) )
             => ( Y
               != ( map_ta8708341958415907829_a_int @ X @ As @ ( cons_int @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_241_map__tailrec__rev_Oelims,axiom,
    ! [X: b > int,Xa: list_b,Xb: list_int,Y: list_int] :
      ( ( ( map_ta720414250517526518_b_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ( Y
               != ( map_ta720414250517526518_b_int @ X @ As @ ( cons_int @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_242_map__tailrec__rev_Oelims,axiom,
    ! [X: nat > nat,Xa: list_nat,Xb: list_nat,Y: list_nat] :
      ( ( ( map_ta7164188454487880599at_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != Xb ) )
       => ~ ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ( Y
               != ( map_ta7164188454487880599at_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_243_map__tailrec__rev_Oelims,axiom,
    ! [X: nat > int,Xa: list_nat,Xb: list_int,Y: list_int] :
      ( ( ( map_ta7161697983978830323at_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_nat )
         => ( Y != Xb ) )
       => ~ ! [A4: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A4 @ As ) )
             => ( Y
               != ( map_ta7161697983978830323at_int @ X @ As @ ( cons_int @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_244_map__tailrec__rev_Oelims,axiom,
    ! [X: int > nat,Xa: list_int,Xb: list_nat,Y: list_nat] :
      ( ( ( map_ta8163542166388935155nt_nat @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ( Y
               != ( map_ta8163542166388935155nt_nat @ X @ As @ ( cons_nat @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_245_map__tailrec__rev_Oelims,axiom,
    ! [X: int > int,Xa: list_int,Xb: list_int,Y: list_int] :
      ( ( ( map_ta8161051695879884879nt_int @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_int )
         => ( Y != Xb ) )
       => ~ ! [A4: int,As: list_int] :
              ( ( Xa
                = ( cons_int @ A4 @ As ) )
             => ( Y
               != ( map_ta8161051695879884879nt_int @ X @ As @ ( cons_int @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_246_map__tailrec__rev_Oelims,axiom,
    ! [X: a > produc6575502325842934193n_assn,Xa: list_a,Xb: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( ( map_ta4951362296667995304n_assn @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_a )
         => ( Y != Xb ) )
       => ~ ! [A4: a,As: list_a] :
              ( ( Xa
                = ( cons_a @ A4 @ As ) )
             => ( Y
               != ( map_ta4951362296667995304n_assn @ X @ As @ ( cons_P2971678138204555879n_assn @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_247_map__tailrec__rev_Oelims,axiom,
    ! [X: b > produc6575502325842934193n_assn,Xa: list_b,Xb: list_P8527749157015355191n_assn,Y: list_P8527749157015355191n_assn] :
      ( ( ( map_ta1524144078346395495n_assn @ X @ Xa @ Xb )
        = Y )
     => ( ( ( Xa = nil_b )
         => ( Y != Xb ) )
       => ~ ! [A4: b,As: list_b] :
              ( ( Xa
                = ( cons_b @ A4 @ As ) )
             => ( Y
               != ( map_ta1524144078346395495n_assn @ X @ As @ ( cons_P2971678138204555879n_assn @ ( X @ A4 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_248_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: a > a > $o,X: a,Xs: list_a] :
      ~ ( lexordp_eq_a @ Less @ ( cons_a @ X @ Xs ) @ nil_a ) ).

% ord.lexordp_eq_simps(3)
thf(fact_249_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: b > b > $o,X: b,Xs: list_b] :
      ~ ( lexordp_eq_b @ Less @ ( cons_b @ X @ Xs ) @ nil_b ) ).

% ord.lexordp_eq_simps(3)
thf(fact_250_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: nat > nat > $o,X: nat,Xs: list_nat] :
      ~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).

% ord.lexordp_eq_simps(3)
thf(fact_251_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: int > int > $o,X: int,Xs: list_int] :
      ~ ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ nil_int ) ).

% ord.lexordp_eq_simps(3)
thf(fact_252_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ~ ( lexord6224210647917505021n_assn @ Less @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ nil_Pr5671120429643327159n_assn ) ).

% ord.lexordp_eq_simps(3)
thf(fact_253_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: nat > nat > $o,X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( Less @ X @ Y )
        | ( ~ ( Less @ Y @ X )
          & ( lexordp_eq_nat @ Less @ Xs @ Ys ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_254_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: int > int > $o,X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( ( Less @ X @ Y )
        | ( ~ ( Less @ Y @ X )
          & ( lexordp_eq_int @ Less @ Xs @ Ys ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_255_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( lexord6224210647917505021n_assn @ Less @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( ( Less @ X @ Y )
        | ( ~ ( Less @ Y @ X )
          & ( lexord6224210647917505021n_assn @ Less @ Xs @ Ys ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_256_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).

% ord.lexordp_eq_simps(1)
thf(fact_257_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: b > b > $o,Ys: list_b] : ( lexordp_eq_b @ Less @ nil_b @ Ys ) ).

% ord.lexordp_eq_simps(1)
thf(fact_258_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Ys: list_P8527749157015355191n_assn] : ( lexord6224210647917505021n_assn @ Less @ nil_Pr5671120429643327159n_assn @ Ys ) ).

% ord.lexordp_eq_simps(1)
thf(fact_259_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).

% ord.lexordp_eq_simps(1)
thf(fact_260_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: int > int > $o,Ys: list_int] : ( lexordp_eq_int @ Less @ nil_int @ Ys ) ).

% ord.lexordp_eq_simps(1)
thf(fact_261_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: a > a > $o,Xs: list_a] :
      ( ( lexordp_eq_a @ Less @ Xs @ nil_a )
      = ( Xs = nil_a ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_262_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: b > b > $o,Xs: list_b] :
      ( ( lexordp_eq_b @ Less @ Xs @ nil_b )
      = ( Xs = nil_b ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_263_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Xs: list_P8527749157015355191n_assn] :
      ( ( lexord6224210647917505021n_assn @ Less @ Xs @ nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_264_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: nat > nat > $o,Xs: list_nat] :
      ( ( lexordp_eq_nat @ Less @ Xs @ nil_nat )
      = ( Xs = nil_nat ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_265_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: int > int > $o,Xs: list_int] :
      ( ( lexordp_eq_int @ Less @ Xs @ nil_int )
      = ( Xs = nil_int ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_266_list__collect__set__simps_I1_J,axiom,
    ! [F: a > set_o] :
      ( ( list_collect_set_a_o @ F @ nil_a )
      = bot_bot_set_o ) ).

% list_collect_set_simps(1)
thf(fact_267_list__collect__set__simps_I1_J,axiom,
    ! [F: b > set_o] :
      ( ( list_collect_set_b_o @ F @ nil_b )
      = bot_bot_set_o ) ).

% list_collect_set_simps(1)
thf(fact_268_list__collect__set__simps_I1_J,axiom,
    ! [F: produc6575502325842934193n_assn > set_o] :
      ( ( list_c5102633440097552245assn_o @ F @ nil_Pr5671120429643327159n_assn )
      = bot_bot_set_o ) ).

% list_collect_set_simps(1)
thf(fact_269_list__collect__set__simps_I1_J,axiom,
    ! [F: nat > set_o] :
      ( ( list_c8047850539171819768_nat_o @ F @ nil_nat )
      = bot_bot_set_o ) ).

% list_collect_set_simps(1)
thf(fact_270_list__collect__set__simps_I1_J,axiom,
    ! [F: int > set_o] :
      ( ( list_c6226808193739131804_int_o @ F @ nil_int )
      = bot_bot_set_o ) ).

% list_collect_set_simps(1)
thf(fact_271_list__collect__set__simps_I1_J,axiom,
    ! [F: a > set_nat] :
      ( ( list_c5512459755930457856_a_nat @ F @ nil_a )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_272_list__collect__set__simps_I1_J,axiom,
    ! [F: b > set_nat] :
      ( ( list_c6747904084886852353_b_nat @ F @ nil_b )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_273_list__collect__set__simps_I1_J,axiom,
    ! [F: produc6575502325842934193n_assn > set_nat] :
      ( ( list_c6061723043370948915sn_nat @ F @ nil_Pr5671120429643327159n_assn )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_274_list__collect__set__simps_I1_J,axiom,
    ! [F: nat > set_nat] :
      ( ( list_c2452340269597857392at_nat @ F @ nil_nat )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_275_list__collect__set__simps_I1_J,axiom,
    ! [F: int > set_nat] :
      ( ( list_c3451693981498911948nt_nat @ F @ nil_int )
      = bot_bot_set_nat ) ).

% list_collect_set_simps(1)
thf(fact_276_ord_Olexordp__eq_OCons,axiom,
    ! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( Less @ X @ Y )
     => ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_277_ord_Olexordp__eq_OCons,axiom,
    ! [Less: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( Less @ X @ Y )
     => ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_278_ord_Olexordp__eq_OCons,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( Less @ X @ Y )
     => ( lexord6224210647917505021n_assn @ Less @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_279_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ ( Less @ X @ Y )
     => ( ~ ( Less @ Y @ X )
       => ( ( lexordp_eq_nat @ Less @ Xs @ Ys )
         => ( lexordp_eq_nat @ Less @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_280_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ ( Less @ X @ Y )
     => ( ~ ( Less @ Y @ X )
       => ( ( lexordp_eq_int @ Less @ Xs @ Ys )
         => ( lexordp_eq_int @ Less @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_281_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ~ ( Less @ X @ Y )
     => ( ~ ( Less @ Y @ X )
       => ( ( lexord6224210647917505021n_assn @ Less @ Xs @ Ys )
         => ( lexord6224210647917505021n_assn @ Less @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_282_ord_Olexordp__eq_ONil,axiom,
    ! [Less: a > a > $o,Ys: list_a] : ( lexordp_eq_a @ Less @ nil_a @ Ys ) ).

% ord.lexordp_eq.Nil
thf(fact_283_ord_Olexordp__eq_ONil,axiom,
    ! [Less: b > b > $o,Ys: list_b] : ( lexordp_eq_b @ Less @ nil_b @ Ys ) ).

% ord.lexordp_eq.Nil
thf(fact_284_ord_Olexordp__eq_ONil,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,Ys: list_P8527749157015355191n_assn] : ( lexord6224210647917505021n_assn @ Less @ nil_Pr5671120429643327159n_assn @ Ys ) ).

% ord.lexordp_eq.Nil
thf(fact_285_ord_Olexordp__eq_ONil,axiom,
    ! [Less: nat > nat > $o,Ys: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys ) ).

% ord.lexordp_eq.Nil
thf(fact_286_ord_Olexordp__eq_ONil,axiom,
    ! [Less: int > int > $o,Ys: list_int] : ( lexordp_eq_int @ Less @ nil_int @ Ys ) ).

% ord.lexordp_eq.Nil
thf(fact_287_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: nat > nat,A: nat,As2: list_nat,Bs: list_nat] :
      ( ( map_ta7164188454487880599at_nat @ F @ ( cons_nat @ A @ As2 ) @ Bs )
      = ( map_ta7164188454487880599at_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_288_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: nat > int,A: nat,As2: list_nat,Bs: list_int] :
      ( ( map_ta7161697983978830323at_int @ F @ ( cons_nat @ A @ As2 ) @ Bs )
      = ( map_ta7161697983978830323at_int @ F @ As2 @ ( cons_int @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_289_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: nat > produc6575502325842934193n_assn,A: nat,As2: list_nat,Bs: list_P8527749157015355191n_assn] :
      ( ( map_ta8369952943552905514n_assn @ F @ ( cons_nat @ A @ As2 ) @ Bs )
      = ( map_ta8369952943552905514n_assn @ F @ As2 @ ( cons_P2971678138204555879n_assn @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_290_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: int > nat,A: int,As2: list_int,Bs: list_nat] :
      ( ( map_ta8163542166388935155nt_nat @ F @ ( cons_int @ A @ As2 ) @ Bs )
      = ( map_ta8163542166388935155nt_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_291_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: int > int,A: int,As2: list_int,Bs: list_int] :
      ( ( map_ta8161051695879884879nt_int @ F @ ( cons_int @ A @ As2 ) @ Bs )
      = ( map_ta8161051695879884879nt_int @ F @ As2 @ ( cons_int @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_292_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: int > produc6575502325842934193n_assn,A: int,As2: list_int,Bs: list_P8527749157015355191n_assn] :
      ( ( map_ta906617570451240910n_assn @ F @ ( cons_int @ A @ As2 ) @ Bs )
      = ( map_ta906617570451240910n_assn @ F @ As2 @ ( cons_P2971678138204555879n_assn @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_293_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: produc6575502325842934193n_assn > nat,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_nat] :
      ( ( map_ta4049178433593862988sn_nat @ F @ ( cons_P2971678138204555879n_assn @ A @ As2 ) @ Bs )
      = ( map_ta4049178433593862988sn_nat @ F @ As2 @ ( cons_nat @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_294_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: produc6575502325842934193n_assn > int,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_int] :
      ( ( map_ta4046687963084812712sn_int @ F @ ( cons_P2971678138204555879n_assn @ A @ As2 ) @ Bs )
      = ( map_ta4046687963084812712sn_int @ F @ As2 @ ( cons_int @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_295_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,Bs: list_P8527749157015355191n_assn] :
      ( ( map_ta6859916920478844725n_assn @ F @ ( cons_P2971678138204555879n_assn @ A @ As2 ) @ Bs )
      = ( map_ta6859916920478844725n_assn @ F @ As2 @ ( cons_P2971678138204555879n_assn @ ( F @ A ) @ Bs ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_296_revg_Osimps_I2_J,axiom,
    ! [A: nat,As2: list_nat,B: list_nat] :
      ( ( revg_nat @ ( cons_nat @ A @ As2 ) @ B )
      = ( revg_nat @ As2 @ ( cons_nat @ A @ B ) ) ) ).

% revg.simps(2)
thf(fact_297_revg_Osimps_I2_J,axiom,
    ! [A: int,As2: list_int,B: list_int] :
      ( ( revg_int @ ( cons_int @ A @ As2 ) @ B )
      = ( revg_int @ As2 @ ( cons_int @ A @ B ) ) ) ).

% revg.simps(2)
thf(fact_298_revg_Osimps_I2_J,axiom,
    ! [A: produc6575502325842934193n_assn,As2: list_P8527749157015355191n_assn,B: list_P8527749157015355191n_assn] :
      ( ( revg_P8856960164974728692n_assn @ ( cons_P2971678138204555879n_assn @ A @ As2 ) @ B )
      = ( revg_P8856960164974728692n_assn @ As2 @ ( cons_P2971678138204555879n_assn @ A @ B ) ) ) ).

% revg.simps(2)
thf(fact_299_revg_Osimps_I1_J,axiom,
    ! [B: list_a] :
      ( ( revg_a @ nil_a @ B )
      = B ) ).

% revg.simps(1)
thf(fact_300_revg_Osimps_I1_J,axiom,
    ! [B: list_b] :
      ( ( revg_b @ nil_b @ B )
      = B ) ).

% revg.simps(1)
thf(fact_301_revg_Osimps_I1_J,axiom,
    ! [B: list_P8527749157015355191n_assn] :
      ( ( revg_P8856960164974728692n_assn @ nil_Pr5671120429643327159n_assn @ B )
      = B ) ).

% revg.simps(1)
thf(fact_302_revg_Osimps_I1_J,axiom,
    ! [B: list_nat] :
      ( ( revg_nat @ nil_nat @ B )
      = B ) ).

% revg.simps(1)
thf(fact_303_revg_Osimps_I1_J,axiom,
    ! [B: list_int] :
      ( ( revg_int @ nil_int @ B )
      = B ) ).

% revg.simps(1)
thf(fact_304_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: a > a > $o,A1: list_a,A22: list_a] :
      ( ( lexordp_eq_a @ Less @ A1 @ A22 )
     => ( ( A1 != nil_a )
       => ( ! [X2: a] :
              ( ? [Xs2: list_a] :
                  ( A1
                  = ( cons_a @ X2 @ Xs2 ) )
             => ! [Y2: a] :
                  ( ? [Ys2: list_a] :
                      ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                 => ~ ( Less @ X2 @ Y2 ) ) )
         => ~ ! [X2: a,Y2: a,Xs2: list_a] :
                ( ( A1
                  = ( cons_a @ X2 @ Xs2 ) )
               => ! [Ys2: list_a] :
                    ( ( A22
                      = ( cons_a @ Y2 @ Ys2 ) )
                   => ( ~ ( Less @ X2 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X2 )
                       => ~ ( lexordp_eq_a @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_305_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: b > b > $o,A1: list_b,A22: list_b] :
      ( ( lexordp_eq_b @ Less @ A1 @ A22 )
     => ( ( A1 != nil_b )
       => ( ! [X2: b] :
              ( ? [Xs2: list_b] :
                  ( A1
                  = ( cons_b @ X2 @ Xs2 ) )
             => ! [Y2: b] :
                  ( ? [Ys2: list_b] :
                      ( A22
                      = ( cons_b @ Y2 @ Ys2 ) )
                 => ~ ( Less @ X2 @ Y2 ) ) )
         => ~ ! [X2: b,Y2: b,Xs2: list_b] :
                ( ( A1
                  = ( cons_b @ X2 @ Xs2 ) )
               => ! [Ys2: list_b] :
                    ( ( A22
                      = ( cons_b @ Y2 @ Ys2 ) )
                   => ( ~ ( Less @ X2 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X2 )
                       => ~ ( lexordp_eq_b @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_306_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: nat > nat > $o,A1: list_nat,A22: list_nat] :
      ( ( lexordp_eq_nat @ Less @ A1 @ A22 )
     => ( ( A1 != nil_nat )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] :
                  ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] :
                      ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                 => ~ ( Less @ X2 @ Y2 ) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
               => ! [Ys2: list_nat] :
                    ( ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                   => ( ~ ( Less @ X2 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X2 )
                       => ~ ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_307_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: int > int > $o,A1: list_int,A22: list_int] :
      ( ( lexordp_eq_int @ Less @ A1 @ A22 )
     => ( ( A1 != nil_int )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] :
                  ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] :
                      ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                 => ~ ( Less @ X2 @ Y2 ) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
               => ! [Ys2: list_int] :
                    ( ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                   => ( ~ ( Less @ X2 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X2 )
                       => ~ ( lexordp_eq_int @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_308_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,A1: list_P8527749157015355191n_assn,A22: list_P8527749157015355191n_assn] :
      ( ( lexord6224210647917505021n_assn @ Less @ A1 @ A22 )
     => ( ( A1 != nil_Pr5671120429643327159n_assn )
       => ( ! [X2: produc6575502325842934193n_assn] :
              ( ? [Xs2: list_P8527749157015355191n_assn] :
                  ( A1
                  = ( cons_P2971678138204555879n_assn @ X2 @ Xs2 ) )
             => ! [Y2: produc6575502325842934193n_assn] :
                  ( ? [Ys2: list_P8527749157015355191n_assn] :
                      ( A22
                      = ( cons_P2971678138204555879n_assn @ Y2 @ Ys2 ) )
                 => ~ ( Less @ X2 @ Y2 ) ) )
         => ~ ! [X2: produc6575502325842934193n_assn,Y2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
                ( ( A1
                  = ( cons_P2971678138204555879n_assn @ X2 @ Xs2 ) )
               => ! [Ys2: list_P8527749157015355191n_assn] :
                    ( ( A22
                      = ( cons_P2971678138204555879n_assn @ Y2 @ Ys2 ) )
                   => ( ~ ( Less @ X2 @ Y2 )
                     => ( ~ ( Less @ Y2 @ X2 )
                       => ~ ( lexord6224210647917505021n_assn @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_309_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_a
    = ( ^ [Less2: a > a > $o,A12: list_a,A23: list_a] :
          ( ? [Ys3: list_a] :
              ( ( A12 = nil_a )
              & ( A23 = Ys3 ) )
          | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_a @ Y3 @ Ys3 ) )
              & ( Less2 @ X3 @ Y3 ) )
          | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_a @ Y3 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y3 )
              & ~ ( Less2 @ Y3 @ X3 )
              & ( lexordp_eq_a @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_310_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_b
    = ( ^ [Less2: b > b > $o,A12: list_b,A23: list_b] :
          ( ? [Ys3: list_b] :
              ( ( A12 = nil_b )
              & ( A23 = Ys3 ) )
          | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_b @ Y3 @ Ys3 ) )
              & ( Less2 @ X3 @ Y3 ) )
          | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_b @ Y3 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y3 )
              & ~ ( Less2 @ Y3 @ X3 )
              & ( lexordp_eq_b @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_311_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_nat
    = ( ^ [Less2: nat > nat > $o,A12: list_nat,A23: list_nat] :
          ( ? [Ys3: list_nat] :
              ( ( A12 = nil_nat )
              & ( A23 = Ys3 ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ( Less2 @ X3 @ Y3 ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y3 )
              & ~ ( Less2 @ Y3 @ X3 )
              & ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_312_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_int
    = ( ^ [Less2: int > int > $o,A12: list_int,A23: list_int] :
          ( ? [Ys3: list_int] :
              ( ( A12 = nil_int )
              & ( A23 = Ys3 ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ( Less2 @ X3 @ Y3 ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y3 )
              & ~ ( Less2 @ Y3 @ X3 )
              & ( lexordp_eq_int @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_313_ord_Olexordp__eq_Osimps,axiom,
    ( lexord6224210647917505021n_assn
    = ( ^ [Less2: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,A12: list_P8527749157015355191n_assn,A23: list_P8527749157015355191n_assn] :
          ( ? [Ys3: list_P8527749157015355191n_assn] :
              ( ( A12 = nil_Pr5671120429643327159n_assn )
              & ( A23 = Ys3 ) )
          | ? [X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs3: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( ( A12
                = ( cons_P2971678138204555879n_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
              & ( Less2 @ X3 @ Y3 ) )
          | ? [X3: produc6575502325842934193n_assn,Y3: produc6575502325842934193n_assn,Xs3: list_P8527749157015355191n_assn,Ys3: list_P8527749157015355191n_assn] :
              ( ( A12
                = ( cons_P2971678138204555879n_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y3 )
              & ~ ( Less2 @ Y3 @ X3 )
              & ( lexord6224210647917505021n_assn @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_314_assn__aci_I10_J,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ ( times_times_assn @ A @ C2 ) @ B ) ) ).

% assn_aci(10)
thf(fact_315_star__aci_I3_J,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) )
      = ( times_times_assn @ B @ ( times_times_assn @ A @ C2 ) ) ) ).

% star_aci(3)
thf(fact_316_star__aci_I2_J,axiom,
    ( times_times_assn
    = ( ^ [A2: assn,B2: assn] : ( times_times_assn @ B2 @ A2 ) ) ) ).

% star_aci(2)
thf(fact_317_star__assoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% star_assoc
thf(fact_318_and__extract__pure__left__iff,axiom,
    ! [B: $o,Q: assn] :
      ( ( inf_inf_assn @ ( pure_assn @ B ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_iff
thf(fact_319_and__extract__pure__right__iff,axiom,
    ! [P: assn,B: $o] :
      ( ( inf_inf_assn @ P @ ( pure_assn @ B ) )
      = ( times_times_assn @ ( inf_inf_assn @ one_one_assn @ P ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_iff
thf(fact_320_listrelp_Ocases,axiom,
    ! [R3: a > a > $o,A1: list_a,A22: list_a] :
      ( ( listrelp_a_a @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_a ) )
       => ~ ! [X2: a,Y2: a,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X2 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_a_a @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_321_listrelp_Ocases,axiom,
    ! [R3: a > b > $o,A1: list_a,A22: list_b] :
      ( ( listrelp_a_b @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_b ) )
       => ~ ! [X2: a,Y2: b,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X2 @ Xs2 ) )
             => ! [Ys2: list_b] :
                  ( ( A22
                    = ( cons_b @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_a_b @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_322_listrelp_Ocases,axiom,
    ! [R3: b > a > $o,A1: list_b,A22: list_a] :
      ( ( listrelp_b_a @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_a ) )
       => ~ ! [X2: b,Y2: a,Xs2: list_b] :
              ( ( A1
                = ( cons_b @ X2 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_b_a @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_323_listrelp_Ocases,axiom,
    ! [R3: b > b > $o,A1: list_b,A22: list_b] :
      ( ( listrelp_b_b @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_b ) )
       => ~ ! [X2: b,Y2: b,Xs2: list_b] :
              ( ( A1
                = ( cons_b @ X2 @ Xs2 ) )
             => ! [Ys2: list_b] :
                  ( ( A22
                    = ( cons_b @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_b_b @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_324_listrelp_Ocases,axiom,
    ! [R3: a > nat > $o,A1: list_a,A22: list_nat] :
      ( ( listrelp_a_nat @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: a,Y2: nat,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X2 @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_a_nat @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_325_listrelp_Ocases,axiom,
    ! [R3: b > nat > $o,A1: list_b,A22: list_nat] :
      ( ( listrelp_b_nat @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: b,Y2: nat,Xs2: list_b] :
              ( ( A1
                = ( cons_b @ X2 @ Xs2 ) )
             => ! [Ys2: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_b_nat @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_326_listrelp_Ocases,axiom,
    ! [R3: a > int > $o,A1: list_a,A22: list_int] :
      ( ( listrelp_a_int @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_int ) )
       => ~ ! [X2: a,Y2: int,Xs2: list_a] :
              ( ( A1
                = ( cons_a @ X2 @ Xs2 ) )
             => ! [Ys2: list_int] :
                  ( ( A22
                    = ( cons_int @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_a_int @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_327_listrelp_Ocases,axiom,
    ! [R3: b > int > $o,A1: list_b,A22: list_int] :
      ( ( listrelp_b_int @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_b )
         => ( A22 != nil_int ) )
       => ~ ! [X2: b,Y2: int,Xs2: list_b] :
              ( ( A1
                = ( cons_b @ X2 @ Xs2 ) )
             => ! [Ys2: list_int] :
                  ( ( A22
                    = ( cons_int @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_b_int @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_328_listrelp_Ocases,axiom,
    ! [R3: nat > a > $o,A1: list_nat,A22: list_a] :
      ( ( listrelp_nat_a @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_a ) )
       => ~ ! [X2: nat,Y2: a,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs2 ) )
             => ! [Ys2: list_a] :
                  ( ( A22
                    = ( cons_a @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_nat_a @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_329_listrelp_Ocases,axiom,
    ! [R3: nat > b > $o,A1: list_nat,A22: list_b] :
      ( ( listrelp_nat_b @ R3 @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_b ) )
       => ~ ! [X2: nat,Y2: b,Xs2: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs2 ) )
             => ! [Ys2: list_b] :
                  ( ( A22
                    = ( cons_b @ Y2 @ Ys2 ) )
                 => ( ( R3 @ X2 @ Y2 )
                   => ~ ( listrelp_nat_b @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_330_listrelp_Osimps,axiom,
    ( listrelp_a_a
    = ( ^ [R4: a > a > $o,A12: list_a,A23: list_a] :
          ( ( ( A12 = nil_a )
            & ( A23 = nil_a ) )
          | ? [X3: a,Y3: a,Xs3: list_a,Ys3: list_a] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_a @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_a_a @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_331_listrelp_Osimps,axiom,
    ( listrelp_a_b
    = ( ^ [R4: a > b > $o,A12: list_a,A23: list_b] :
          ( ( ( A12 = nil_a )
            & ( A23 = nil_b ) )
          | ? [X3: a,Y3: b,Xs3: list_a,Ys3: list_b] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_b @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_a_b @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_332_listrelp_Osimps,axiom,
    ( listrelp_b_a
    = ( ^ [R4: b > a > $o,A12: list_b,A23: list_a] :
          ( ( ( A12 = nil_b )
            & ( A23 = nil_a ) )
          | ? [X3: b,Y3: a,Xs3: list_b,Ys3: list_a] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_a @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_b_a @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_333_listrelp_Osimps,axiom,
    ( listrelp_b_b
    = ( ^ [R4: b > b > $o,A12: list_b,A23: list_b] :
          ( ( ( A12 = nil_b )
            & ( A23 = nil_b ) )
          | ? [X3: b,Y3: b,Xs3: list_b,Ys3: list_b] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_b @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_b_b @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_334_listrelp_Osimps,axiom,
    ( listrelp_a_nat
    = ( ^ [R4: a > nat > $o,A12: list_a,A23: list_nat] :
          ( ( ( A12 = nil_a )
            & ( A23 = nil_nat ) )
          | ? [X3: a,Y3: nat,Xs3: list_a,Ys3: list_nat] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_a_nat @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_335_listrelp_Osimps,axiom,
    ( listrelp_b_nat
    = ( ^ [R4: b > nat > $o,A12: list_b,A23: list_nat] :
          ( ( ( A12 = nil_b )
            & ( A23 = nil_nat ) )
          | ? [X3: b,Y3: nat,Xs3: list_b,Ys3: list_nat] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_b_nat @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_336_listrelp_Osimps,axiom,
    ( listrelp_a_int
    = ( ^ [R4: a > int > $o,A12: list_a,A23: list_int] :
          ( ( ( A12 = nil_a )
            & ( A23 = nil_int ) )
          | ? [X3: a,Y3: int,Xs3: list_a,Ys3: list_int] :
              ( ( A12
                = ( cons_a @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_a_int @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_337_listrelp_Osimps,axiom,
    ( listrelp_b_int
    = ( ^ [R4: b > int > $o,A12: list_b,A23: list_int] :
          ( ( ( A12 = nil_b )
            & ( A23 = nil_int ) )
          | ? [X3: b,Y3: int,Xs3: list_b,Ys3: list_int] :
              ( ( A12
                = ( cons_b @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_b_int @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_338_listrelp_Osimps,axiom,
    ( listrelp_nat_a
    = ( ^ [R4: nat > a > $o,A12: list_nat,A23: list_a] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_a ) )
          | ? [X3: nat,Y3: a,Xs3: list_nat,Ys3: list_a] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_a @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_nat_a @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_339_listrelp_Osimps,axiom,
    ( listrelp_nat_b
    = ( ^ [R4: nat > b > $o,A12: list_nat,A23: list_b] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_b ) )
          | ? [X3: nat,Y3: b,Xs3: list_nat,Ys3: list_b] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_b @ Y3 @ Ys3 ) )
              & ( R4 @ X3 @ Y3 )
              & ( listrelp_nat_b @ R4 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_340_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: a > a > $o,X: a] :
      ( ( mergesort_by_rel_a @ R @ ( cons_a @ X @ nil_a ) )
      = ( cons_a @ X @ nil_a ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_341_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: b > b > $o,X: b] :
      ( ( mergesort_by_rel_b @ R @ ( cons_b @ X @ nil_b ) )
      = ( cons_b @ X @ nil_b ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_342_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: nat > nat > $o,X: nat] :
      ( ( mergesort_by_rel_nat @ R @ ( cons_nat @ X @ nil_nat ) )
      = ( cons_nat @ X @ nil_nat ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_343_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: int > int > $o,X: int] :
      ( ( mergesort_by_rel_int @ R @ ( cons_int @ X @ nil_int ) )
      = ( cons_int @ X @ nil_int ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_344_mergesort__by__rel__simps_I2_J,axiom,
    ! [R: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn] :
      ( ( merges5609009838848816300n_assn @ R @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ).

% mergesort_by_rel_simps(2)
thf(fact_345_lexordp__eq__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ~ ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ nil_nat ) ).

% lexordp_eq_simps(3)
thf(fact_346_lexordp__eq__simps_I3_J,axiom,
    ! [X: int,Xs: list_int] :
      ~ ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ nil_int ) ).

% lexordp_eq_simps(3)
thf(fact_347_entails__solve__finalize_I2_J,axiom,
    ! [M: list_P8527749157015355191n_assn] : ( fI_RESULT @ M @ one_one_assn @ one_one_assn @ one_one_assn ) ).

% entails_solve_finalize(2)
thf(fact_348_frame__inference__finalize,axiom,
    ! [M: list_P8527749157015355191n_assn,F2: assn] : ( fI_RESULT @ M @ F2 @ one_one_assn @ F2 ) ).

% frame_inference_finalize
thf(fact_349_mult_Oright__assoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ A @ ( times_times_assn @ B @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_350_mult_Oright__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% mult.right_assoc
thf(fact_351_mult_Oright__commute,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( times_times_assn @ ( times_times_assn @ A @ B ) @ C2 )
      = ( times_times_assn @ ( times_times_assn @ A @ C2 ) @ B ) ) ).

% mult.right_commute
thf(fact_352_mult_Oright__commute,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ).

% mult.right_commute
thf(fact_353_list_Omap__disc__iff,axiom,
    ! [F: a > a,A: list_a] :
      ( ( ( map_a_a @ F @ A )
        = nil_a )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_354_list_Omap__disc__iff,axiom,
    ! [F: b > a,A: list_b] :
      ( ( ( map_b_a @ F @ A )
        = nil_a )
      = ( A = nil_b ) ) ).

% list.map_disc_iff
thf(fact_355_list_Omap__disc__iff,axiom,
    ! [F: nat > a,A: list_nat] :
      ( ( ( map_nat_a @ F @ A )
        = nil_a )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_356_list_Omap__disc__iff,axiom,
    ! [F: int > a,A: list_int] :
      ( ( ( map_int_a @ F @ A )
        = nil_a )
      = ( A = nil_int ) ) ).

% list.map_disc_iff
thf(fact_357_list_Omap__disc__iff,axiom,
    ! [F: a > b,A: list_a] :
      ( ( ( map_a_b @ F @ A )
        = nil_b )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_358_list_Omap__disc__iff,axiom,
    ! [F: b > b,A: list_b] :
      ( ( ( map_b_b @ F @ A )
        = nil_b )
      = ( A = nil_b ) ) ).

% list.map_disc_iff
thf(fact_359_list_Omap__disc__iff,axiom,
    ! [F: nat > b,A: list_nat] :
      ( ( ( map_nat_b @ F @ A )
        = nil_b )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_360_list_Omap__disc__iff,axiom,
    ! [F: int > b,A: list_int] :
      ( ( ( map_int_b @ F @ A )
        = nil_b )
      = ( A = nil_int ) ) ).

% list.map_disc_iff
thf(fact_361_list_Omap__disc__iff,axiom,
    ! [F: a > nat,A: list_a] :
      ( ( ( map_a_nat @ F @ A )
        = nil_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_362_list_Omap__disc__iff,axiom,
    ! [F: b > nat,A: list_b] :
      ( ( ( map_b_nat @ F @ A )
        = nil_nat )
      = ( A = nil_b ) ) ).

% list.map_disc_iff
thf(fact_363_Nil__is__map__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( nil_a
        = ( map_a_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_364_Nil__is__map__conv,axiom,
    ! [F: b > a,Xs: list_b] :
      ( ( nil_a
        = ( map_b_a @ F @ Xs ) )
      = ( Xs = nil_b ) ) ).

% Nil_is_map_conv
thf(fact_365_Nil__is__map__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( nil_a
        = ( map_nat_a @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_366_Nil__is__map__conv,axiom,
    ! [F: int > a,Xs: list_int] :
      ( ( nil_a
        = ( map_int_a @ F @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_map_conv
thf(fact_367_Nil__is__map__conv,axiom,
    ! [F: a > b,Xs: list_a] :
      ( ( nil_b
        = ( map_a_b @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_368_Nil__is__map__conv,axiom,
    ! [F: b > b,Xs: list_b] :
      ( ( nil_b
        = ( map_b_b @ F @ Xs ) )
      = ( Xs = nil_b ) ) ).

% Nil_is_map_conv
thf(fact_369_Nil__is__map__conv,axiom,
    ! [F: nat > b,Xs: list_nat] :
      ( ( nil_b
        = ( map_nat_b @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_370_Nil__is__map__conv,axiom,
    ! [F: int > b,Xs: list_int] :
      ( ( nil_b
        = ( map_int_b @ F @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_map_conv
thf(fact_371_Nil__is__map__conv,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( nil_nat
        = ( map_a_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_372_Nil__is__map__conv,axiom,
    ! [F: b > nat,Xs: list_b] :
      ( ( nil_nat
        = ( map_b_nat @ F @ Xs ) )
      = ( Xs = nil_b ) ) ).

% Nil_is_map_conv
thf(fact_373_map__is__Nil__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_374_map__is__Nil__conv,axiom,
    ! [F: b > a,Xs: list_b] :
      ( ( ( map_b_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_b ) ) ).

% map_is_Nil_conv
thf(fact_375_map__is__Nil__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_376_map__is__Nil__conv,axiom,
    ! [F: int > a,Xs: list_int] :
      ( ( ( map_int_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_int ) ) ).

% map_is_Nil_conv
thf(fact_377_map__is__Nil__conv,axiom,
    ! [F: a > b,Xs: list_a] :
      ( ( ( map_a_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_378_map__is__Nil__conv,axiom,
    ! [F: b > b,Xs: list_b] :
      ( ( ( map_b_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_b ) ) ).

% map_is_Nil_conv
thf(fact_379_map__is__Nil__conv,axiom,
    ! [F: nat > b,Xs: list_nat] :
      ( ( ( map_nat_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_380_map__is__Nil__conv,axiom,
    ! [F: int > b,Xs: list_int] :
      ( ( ( map_int_b @ F @ Xs )
        = nil_b )
      = ( Xs = nil_int ) ) ).

% map_is_Nil_conv
thf(fact_381_map__is__Nil__conv,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( ( map_a_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_382_map__is__Nil__conv,axiom,
    ! [F: b > nat,Xs: list_b] :
      ( ( ( map_b_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_b ) ) ).

% map_is_Nil_conv
thf(fact_383_lexordp__eq__simps_I2_J,axiom,
    ! [Xs: list_nat] :
      ( ( ord_lexordp_eq_nat @ Xs @ nil_nat )
      = ( Xs = nil_nat ) ) ).

% lexordp_eq_simps(2)
thf(fact_384_lexordp__eq__simps_I2_J,axiom,
    ! [Xs: list_int] :
      ( ( ord_lexordp_eq_int @ Xs @ nil_int )
      = ( Xs = nil_int ) ) ).

% lexordp_eq_simps(2)
thf(fact_385_lexordp__eq__simps_I1_J,axiom,
    ! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).

% lexordp_eq_simps(1)
thf(fact_386_lexordp__eq__simps_I1_J,axiom,
    ! [Ys: list_int] : ( ord_lexordp_eq_int @ nil_int @ Ys ) ).

% lexordp_eq_simps(1)
thf(fact_387_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: a > a > $o] :
      ( ( mergesort_by_rel_a @ R @ nil_a )
      = nil_a ) ).

% mergesort_by_rel_simps(1)
thf(fact_388_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: b > b > $o] :
      ( ( mergesort_by_rel_b @ R @ nil_b )
      = nil_b ) ).

% mergesort_by_rel_simps(1)
thf(fact_389_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] :
      ( ( merges5609009838848816300n_assn @ R @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% mergesort_by_rel_simps(1)
thf(fact_390_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: nat > nat > $o] :
      ( ( mergesort_by_rel_nat @ R @ nil_nat )
      = nil_nat ) ).

% mergesort_by_rel_simps(1)
thf(fact_391_mergesort__by__rel__simps_I1_J,axiom,
    ! [R: int > int > $o] :
      ( ( mergesort_by_rel_int @ R @ nil_int )
      = nil_int ) ).

% mergesort_by_rel_simps(1)
thf(fact_392_merge__pure__and,axiom,
    ! [A: $o,B: $o] :
      ( ( inf_inf_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          & B ) ) ) ).

% merge_pure_and
thf(fact_393_and__extract__pure__right__ctx__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( inf_inf_assn @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_right_ctx_iff
thf(fact_394_and__extract__pure__left__ctx__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( inf_inf_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( times_times_assn @ ( inf_inf_assn @ P @ Q ) @ ( pure_assn @ B ) ) ) ).

% and_extract_pure_left_ctx_iff
thf(fact_395_list__collect__set__map__simps_I1_J,axiom,
    ! [F: nat > set_o,X: nat > nat] :
      ( ( list_c8047850539171819768_nat_o @ F @ ( map_nat_nat @ X @ nil_nat ) )
      = bot_bot_set_o ) ).

% list_collect_set_map_simps(1)
thf(fact_396_list__collect__set__map__simps_I1_J,axiom,
    ! [F: assn > set_o,X: produc6575502325842934193n_assn > assn] :
      ( ( list_c312183563312650144assn_o @ F @ ( map_Pr8991440229025900053n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = bot_bot_set_o ) ).

% list_collect_set_map_simps(1)
thf(fact_397_list__collect__set__map__simps_I1_J,axiom,
    ! [F: nat > set_nat,X: nat > nat] :
      ( ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ nil_nat ) )
      = bot_bot_set_nat ) ).

% list_collect_set_map_simps(1)
thf(fact_398_list__collect__set__map__simps_I1_J,axiom,
    ! [F: assn > set_nat,X: produc6575502325842934193n_assn > assn] :
      ( ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = bot_bot_set_nat ) ).

% list_collect_set_map_simps(1)
thf(fact_399_assn__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% assn_aci(4)
thf(fact_400_assn__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% assn_aci(3)
thf(fact_401_assn__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X3: assn,Y3: assn] : ( inf_inf_assn @ Y3 @ X3 ) ) ) ).

% assn_aci(1)
thf(fact_402_norm__assertion__simps_I31_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(31)
thf(fact_403_norm__assertion__simps_I14_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(14)
thf(fact_404_memb__imp__not__empty,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( S != bot_bo7824918357723582233_nat_o ) ) ).

% memb_imp_not_empty
thf(fact_405_memb__imp__not__empty,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( S != bot_bot_set_o ) ) ).

% memb_imp_not_empty
thf(fact_406_memb__imp__not__empty,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat2 @ X @ S )
     => ( S != bot_bot_set_nat ) ) ).

% memb_imp_not_empty
thf(fact_407_set__notEmptyE,axiom,
    ! [S: set_Pr4532377907799695533_nat_o] :
      ( ( S != bot_bo7824918357723582233_nat_o )
     => ~ ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X2 @ S ) ) ).

% set_notEmptyE
thf(fact_408_set__notEmptyE,axiom,
    ! [S: set_o] :
      ( ( S != bot_bot_set_o )
     => ~ ! [X2: $o] :
            ~ ( member_o @ X2 @ S ) ) ).

% set_notEmptyE
thf(fact_409_set__notEmptyE,axiom,
    ! [S: set_nat] :
      ( ( S != bot_bot_set_nat )
     => ~ ! [X2: nat] :
            ~ ( member_nat2 @ X2 @ S ) ) ).

% set_notEmptyE
thf(fact_410_map__eq__consE,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Ls: list_P8527749157015355191n_assn,Fa: assn,Fl: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Ls )
        = ( cons_assn @ Fa @ Fl ) )
     => ~ ! [A4: produc6575502325842934193n_assn,L2: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( cons_P2971678138204555879n_assn @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_Pr8991440229025900053n_assn @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_411_map__eq__consE,axiom,
    ! [F: nat > nat,Ls: list_nat,Fa: nat,Fl: list_nat] :
      ( ( ( map_nat_nat @ F @ Ls )
        = ( cons_nat @ Fa @ Fl ) )
     => ~ ! [A4: nat,L2: list_nat] :
            ( ( Ls
              = ( cons_nat @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_nat_nat @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_412_map__eq__consE,axiom,
    ! [F: int > nat,Ls: list_int,Fa: nat,Fl: list_nat] :
      ( ( ( map_int_nat @ F @ Ls )
        = ( cons_nat @ Fa @ Fl ) )
     => ~ ! [A4: int,L2: list_int] :
            ( ( Ls
              = ( cons_int @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_int_nat @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_413_map__eq__consE,axiom,
    ! [F: produc6575502325842934193n_assn > nat,Ls: list_P8527749157015355191n_assn,Fa: nat,Fl: list_nat] :
      ( ( ( map_Pr7570552894071451325sn_nat @ F @ Ls )
        = ( cons_nat @ Fa @ Fl ) )
     => ~ ! [A4: produc6575502325842934193n_assn,L2: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( cons_P2971678138204555879n_assn @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_Pr7570552894071451325sn_nat @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_414_map__eq__consE,axiom,
    ! [F: nat > int,Ls: list_nat,Fa: int,Fl: list_int] :
      ( ( ( map_nat_int @ F @ Ls )
        = ( cons_int @ Fa @ Fl ) )
     => ~ ! [A4: nat,L2: list_nat] :
            ( ( Ls
              = ( cons_nat @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_nat_int @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_415_map__eq__consE,axiom,
    ! [F: int > int,Ls: list_int,Fa: int,Fl: list_int] :
      ( ( ( map_int_int @ F @ Ls )
        = ( cons_int @ Fa @ Fl ) )
     => ~ ! [A4: int,L2: list_int] :
            ( ( Ls
              = ( cons_int @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_int_int @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_416_map__eq__consE,axiom,
    ! [F: produc6575502325842934193n_assn > int,Ls: list_P8527749157015355191n_assn,Fa: int,Fl: list_int] :
      ( ( ( map_Pr7568062423562401049sn_int @ F @ Ls )
        = ( cons_int @ Fa @ Fl ) )
     => ~ ! [A4: produc6575502325842934193n_assn,L2: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( cons_P2971678138204555879n_assn @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_Pr7568062423562401049sn_int @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_417_map__eq__consE,axiom,
    ! [F: nat > produc6575502325842934193n_assn,Ls: list_nat,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( ( map_na2667955367175718043n_assn @ F @ Ls )
        = ( cons_P2971678138204555879n_assn @ Fa @ Fl ) )
     => ~ ! [A4: nat,L2: list_nat] :
            ( ( Ls
              = ( cons_nat @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_na2667955367175718043n_assn @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_418_map__eq__consE,axiom,
    ! [F: int > produc6575502325842934193n_assn,Ls: list_int,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( ( map_in4427992030928829247n_assn @ F @ Ls )
        = ( cons_P2971678138204555879n_assn @ Fa @ Fl ) )
     => ~ ! [A4: int,L2: list_int] :
            ( ( Ls
              = ( cons_int @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_in4427992030928829247n_assn @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_419_map__eq__consE,axiom,
    ! [F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Ls: list_P8527749157015355191n_assn,Fa: produc6575502325842934193n_assn,Fl: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7925354932063753860n_assn @ F @ Ls )
        = ( cons_P2971678138204555879n_assn @ Fa @ Fl ) )
     => ~ ! [A4: produc6575502325842934193n_assn,L2: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( cons_P2971678138204555879n_assn @ A4 @ L2 ) )
           => ( ( ( F @ A4 )
                = Fa )
             => ( ( map_Pr7925354932063753860n_assn @ F @ L2 )
               != Fl ) ) ) ) ).

% map_eq_consE
thf(fact_420_map__consI_I1_J,axiom,
    ! [W: list_assn,F: produc6575502325842934193n_assn > assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W
        = ( map_Pr8991440229025900053n_assn @ F @ Ww ) )
     => ( ( cons_assn @ ( F @ A ) @ W )
        = ( map_Pr8991440229025900053n_assn @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_421_map__consI_I1_J,axiom,
    ! [W: list_nat,F: nat > nat,Ww: list_nat,A: nat] :
      ( ( W
        = ( map_nat_nat @ F @ Ww ) )
     => ( ( cons_nat @ ( F @ A ) @ W )
        = ( map_nat_nat @ F @ ( cons_nat @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_422_map__consI_I1_J,axiom,
    ! [W: list_nat,F: int > nat,Ww: list_int,A: int] :
      ( ( W
        = ( map_int_nat @ F @ Ww ) )
     => ( ( cons_nat @ ( F @ A ) @ W )
        = ( map_int_nat @ F @ ( cons_int @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_423_map__consI_I1_J,axiom,
    ! [W: list_nat,F: produc6575502325842934193n_assn > nat,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W
        = ( map_Pr7570552894071451325sn_nat @ F @ Ww ) )
     => ( ( cons_nat @ ( F @ A ) @ W )
        = ( map_Pr7570552894071451325sn_nat @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_424_map__consI_I1_J,axiom,
    ! [W: list_int,F: nat > int,Ww: list_nat,A: nat] :
      ( ( W
        = ( map_nat_int @ F @ Ww ) )
     => ( ( cons_int @ ( F @ A ) @ W )
        = ( map_nat_int @ F @ ( cons_nat @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_425_map__consI_I1_J,axiom,
    ! [W: list_int,F: int > int,Ww: list_int,A: int] :
      ( ( W
        = ( map_int_int @ F @ Ww ) )
     => ( ( cons_int @ ( F @ A ) @ W )
        = ( map_int_int @ F @ ( cons_int @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_426_map__consI_I1_J,axiom,
    ! [W: list_int,F: produc6575502325842934193n_assn > int,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W
        = ( map_Pr7568062423562401049sn_int @ F @ Ww ) )
     => ( ( cons_int @ ( F @ A ) @ W )
        = ( map_Pr7568062423562401049sn_int @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_427_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: nat > produc6575502325842934193n_assn,Ww: list_nat,A: nat] :
      ( ( W
        = ( map_na2667955367175718043n_assn @ F @ Ww ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ W )
        = ( map_na2667955367175718043n_assn @ F @ ( cons_nat @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_428_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: int > produc6575502325842934193n_assn,Ww: list_int,A: int] :
      ( ( W
        = ( map_in4427992030928829247n_assn @ F @ Ww ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ W )
        = ( map_in4427992030928829247n_assn @ F @ ( cons_int @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_429_map__consI_I1_J,axiom,
    ! [W: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( W
        = ( map_Pr7925354932063753860n_assn @ F @ Ww ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ W )
        = ( map_Pr7925354932063753860n_assn @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) ) ) ).

% map_consI(1)
thf(fact_430_map__eq__Cons__conv,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( cons_assn @ Y @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_Pr8991440229025900053n_assn @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_431_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_nat_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_432_map__eq__Cons__conv,axiom,
    ! [F: int > nat,Xs: list_int,Y: nat,Ys: list_nat] :
      ( ( ( map_int_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Xs
              = ( cons_int @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_int_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_433_map__eq__Cons__conv,axiom,
    ! [F: produc6575502325842934193n_assn > nat,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] :
      ( ( ( map_Pr7570552894071451325sn_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_Pr7570552894071451325sn_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_434_map__eq__Cons__conv,axiom,
    ! [F: nat > int,Xs: list_nat,Y: int,Ys: list_int] :
      ( ( ( map_nat_int @ F @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_nat_int @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_435_map__eq__Cons__conv,axiom,
    ! [F: int > int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( ( map_int_int @ F @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Xs
              = ( cons_int @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_int_int @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_436_map__eq__Cons__conv,axiom,
    ! [F: produc6575502325842934193n_assn > int,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] :
      ( ( ( map_Pr7568062423562401049sn_int @ F @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_Pr7568062423562401049sn_int @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_437_map__eq__Cons__conv,axiom,
    ! [F: nat > produc6575502325842934193n_assn,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_na2667955367175718043n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_na2667955367175718043n_assn @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_438_map__eq__Cons__conv,axiom,
    ! [F: int > produc6575502325842934193n_assn,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_in4427992030928829247n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Xs
              = ( cons_int @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_in4427992030928829247n_assn @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_439_map__eq__Cons__conv,axiom,
    ! [F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7925354932063753860n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( ( F @ Z2 )
              = Y )
            & ( ( map_Pr7925354932063753860n_assn @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_440_Cons__eq__map__conv,axiom,
    ! [X: assn,Xs: list_assn,F: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_assn @ X @ Xs )
        = ( map_Pr8991440229025900053n_assn @ F @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_Pr8991440229025900053n_assn @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_441_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_442_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: int > nat,Ys: list_int] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_int_nat @ F @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Ys
              = ( cons_int @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_int_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_443_Cons__eq__map__conv,axiom,
    ! [X: nat,Xs: list_nat,F: produc6575502325842934193n_assn > nat,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_Pr7570552894071451325sn_nat @ F @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_Pr7570552894071451325sn_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_444_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: nat > int,Ys: list_nat] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_nat_int @ F @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_nat_int @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_445_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: int > int,Ys: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_int_int @ F @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Ys
              = ( cons_int @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_int_int @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_446_Cons__eq__map__conv,axiom,
    ! [X: int,Xs: list_int,F: produc6575502325842934193n_assn > int,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_Pr7568062423562401049sn_int @ F @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_Pr7568062423562401049sn_int @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_447_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: nat > produc6575502325842934193n_assn,Ys: list_nat] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_na2667955367175718043n_assn @ F @ Ys ) )
      = ( ? [Z2: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_na2667955367175718043n_assn @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_448_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: int > produc6575502325842934193n_assn,Ys: list_int] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_in4427992030928829247n_assn @ F @ Ys ) )
      = ( ? [Z2: int,Zs: list_int] :
            ( ( Ys
              = ( cons_int @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_in4427992030928829247n_assn @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_449_Cons__eq__map__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_Pr7925354932063753860n_assn @ F @ Ys ) )
      = ( ? [Z2: produc6575502325842934193n_assn,Zs: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ Z2 @ Zs ) )
            & ( X
              = ( F @ Z2 ) )
            & ( Xs
              = ( map_Pr7925354932063753860n_assn @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_450_map__eq__Cons__D,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Y: assn,Ys: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( cons_assn @ Y @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr8991440229025900053n_assn @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_451_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_452_map__eq__Cons__D,axiom,
    ! [F: 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 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_int_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_453_map__eq__Cons__D,axiom,
    ! [F: produc6575502325842934193n_assn > nat,Xs: list_P8527749157015355191n_assn,Y: nat,Ys: list_nat] :
      ( ( ( map_Pr7570552894071451325sn_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr7570552894071451325sn_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_454_map__eq__Cons__D,axiom,
    ! [F: 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 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_nat_int @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_455_map__eq__Cons__D,axiom,
    ! [F: 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 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_int_int @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_456_map__eq__Cons__D,axiom,
    ! [F: produc6575502325842934193n_assn > int,Xs: list_P8527749157015355191n_assn,Y: int,Ys: list_int] :
      ( ( ( map_Pr7568062423562401049sn_int @ F @ Xs )
        = ( cons_int @ Y @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr7568062423562401049sn_int @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_457_map__eq__Cons__D,axiom,
    ! [F: nat > produc6575502325842934193n_assn,Xs: list_nat,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_na2667955367175718043n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_na2667955367175718043n_assn @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_458_map__eq__Cons__D,axiom,
    ! [F: int > produc6575502325842934193n_assn,Xs: list_int,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_in4427992030928829247n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Xs
            = ( cons_int @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_in4427992030928829247n_assn @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_459_map__eq__Cons__D,axiom,
    ! [F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr7925354932063753860n_assn @ F @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Xs
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr7925354932063753860n_assn @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_460_Cons__eq__map__D,axiom,
    ! [X: assn,Xs: list_assn,F: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_assn @ X @ Xs )
        = ( map_Pr8991440229025900053n_assn @ F @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr8991440229025900053n_assn @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_461_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_462_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: 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
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_int_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_463_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: produc6575502325842934193n_assn > nat,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_Pr7570552894071451325sn_nat @ F @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr7570552894071451325sn_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_464_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: 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
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_int @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_465_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: 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
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_int_int @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_466_Cons__eq__map__D,axiom,
    ! [X: int,Xs: list_int,F: produc6575502325842934193n_assn > int,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_int @ X @ Xs )
        = ( map_Pr7568062423562401049sn_int @ F @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr7568062423562401049sn_int @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_467_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: nat > produc6575502325842934193n_assn,Ys: list_nat] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_na2667955367175718043n_assn @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na2667955367175718043n_assn @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_468_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: int > produc6575502325842934193n_assn,Ys: list_int] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_in4427992030928829247n_assn @ F @ Ys ) )
     => ? [Z3: int,Zs2: list_int] :
          ( ( Ys
            = ( cons_int @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_in4427992030928829247n_assn @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_469_Cons__eq__map__D,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( map_Pr7925354932063753860n_assn @ F @ Ys ) )
     => ? [Z3: produc6575502325842934193n_assn,Zs2: list_P8527749157015355191n_assn] :
          ( ( Ys
            = ( cons_P2971678138204555879n_assn @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr7925354932063753860n_assn @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_470_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X22: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_471_list_Osimps_I9_J,axiom,
    ! [F: nat > int,X21: nat,X22: list_nat] :
      ( ( map_nat_int @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_int @ ( F @ X21 ) @ ( map_nat_int @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_472_list_Osimps_I9_J,axiom,
    ! [F: nat > produc6575502325842934193n_assn,X21: nat,X22: list_nat] :
      ( ( map_na2667955367175718043n_assn @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_P2971678138204555879n_assn @ ( F @ X21 ) @ ( map_na2667955367175718043n_assn @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_473_list_Osimps_I9_J,axiom,
    ! [F: int > nat,X21: int,X22: list_int] :
      ( ( map_int_nat @ F @ ( cons_int @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_int_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_474_list_Osimps_I9_J,axiom,
    ! [F: int > int,X21: int,X22: list_int] :
      ( ( map_int_int @ F @ ( cons_int @ X21 @ X22 ) )
      = ( cons_int @ ( F @ X21 ) @ ( map_int_int @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_475_list_Osimps_I9_J,axiom,
    ! [F: int > produc6575502325842934193n_assn,X21: int,X22: list_int] :
      ( ( map_in4427992030928829247n_assn @ F @ ( cons_int @ X21 @ X22 ) )
      = ( cons_P2971678138204555879n_assn @ ( F @ X21 ) @ ( map_in4427992030928829247n_assn @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_476_list_Osimps_I9_J,axiom,
    ! [F: produc6575502325842934193n_assn > assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn @ F @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( cons_assn @ ( F @ X21 ) @ ( map_Pr8991440229025900053n_assn @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_477_list_Osimps_I9_J,axiom,
    ! [F: produc6575502325842934193n_assn > nat,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( map_Pr7570552894071451325sn_nat @ F @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Pr7570552894071451325sn_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_478_list_Osimps_I9_J,axiom,
    ! [F: produc6575502325842934193n_assn > int,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( map_Pr7568062423562401049sn_int @ F @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( cons_int @ ( F @ X21 ) @ ( map_Pr7568062423562401049sn_int @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_479_list_Osimps_I9_J,axiom,
    ! [F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( map_Pr7925354932063753860n_assn @ F @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( cons_P2971678138204555879n_assn @ ( F @ X21 ) @ ( map_Pr7925354932063753860n_assn @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_480_list_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_a_a @ F @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_481_list_Osimps_I8_J,axiom,
    ! [F: a > b] :
      ( ( map_a_b @ F @ nil_a )
      = nil_b ) ).

% list.simps(8)
thf(fact_482_list_Osimps_I8_J,axiom,
    ! [F: a > nat] :
      ( ( map_a_nat @ F @ nil_a )
      = nil_nat ) ).

% list.simps(8)
thf(fact_483_list_Osimps_I8_J,axiom,
    ! [F: a > int] :
      ( ( map_a_int @ F @ nil_a )
      = nil_int ) ).

% list.simps(8)
thf(fact_484_list_Osimps_I8_J,axiom,
    ! [F: b > a] :
      ( ( map_b_a @ F @ nil_b )
      = nil_a ) ).

% list.simps(8)
thf(fact_485_list_Osimps_I8_J,axiom,
    ! [F: b > b] :
      ( ( map_b_b @ F @ nil_b )
      = nil_b ) ).

% list.simps(8)
thf(fact_486_list_Osimps_I8_J,axiom,
    ! [F: b > nat] :
      ( ( map_b_nat @ F @ nil_b )
      = nil_nat ) ).

% list.simps(8)
thf(fact_487_list_Osimps_I8_J,axiom,
    ! [F: b > int] :
      ( ( map_b_int @ F @ nil_b )
      = nil_int ) ).

% list.simps(8)
thf(fact_488_list_Osimps_I8_J,axiom,
    ! [F: nat > a] :
      ( ( map_nat_a @ F @ nil_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_489_list_Osimps_I8_J,axiom,
    ! [F: nat > b] :
      ( ( map_nat_b @ F @ nil_nat )
      = nil_b ) ).

% list.simps(8)
thf(fact_490_norm__assertion__simps_I9_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% norm_assertion_simps(9)
thf(fact_491_norm__assertion__simps_I10_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% norm_assertion_simps(10)
thf(fact_492_lexordp__eq_ONil,axiom,
    ! [Ys: list_nat] : ( ord_lexordp_eq_nat @ nil_nat @ Ys ) ).

% lexordp_eq.Nil
thf(fact_493_lexordp__eq_ONil,axiom,
    ! [Ys: list_int] : ( ord_lexordp_eq_int @ nil_int @ Ys ) ).

% lexordp_eq.Nil
thf(fact_494_listrelp_OCons,axiom,
    ! [R3: nat > nat > $o,X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( R3 @ X @ Y )
     => ( ( listrelp_nat_nat @ R3 @ Xs @ Ys )
       => ( listrelp_nat_nat @ R3 @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_495_listrelp_OCons,axiom,
    ! [R3: nat > int > $o,X: nat,Y: int,Xs: list_nat,Ys: list_int] :
      ( ( R3 @ X @ Y )
     => ( ( listrelp_nat_int @ R3 @ Xs @ Ys )
       => ( listrelp_nat_int @ R3 @ ( cons_nat @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_496_listrelp_OCons,axiom,
    ! [R3: nat > produc6575502325842934193n_assn > $o,X: nat,Y: produc6575502325842934193n_assn,Xs: list_nat,Ys: list_P8527749157015355191n_assn] :
      ( ( R3 @ X @ Y )
     => ( ( listre8980241315090511200n_assn @ R3 @ Xs @ Ys )
       => ( listre8980241315090511200n_assn @ R3 @ ( cons_nat @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_497_listrelp_OCons,axiom,
    ! [R3: int > nat > $o,X: int,Y: nat,Xs: list_int,Ys: list_nat] :
      ( ( R3 @ X @ Y )
     => ( ( listrelp_int_nat @ R3 @ Xs @ Ys )
       => ( listrelp_int_nat @ R3 @ ( cons_int @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_498_listrelp_OCons,axiom,
    ! [R3: int > int > $o,X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( R3 @ X @ Y )
     => ( ( listrelp_int_int @ R3 @ Xs @ Ys )
       => ( listrelp_int_int @ R3 @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_499_listrelp_OCons,axiom,
    ! [R3: int > produc6575502325842934193n_assn > $o,X: int,Y: produc6575502325842934193n_assn,Xs: list_int,Ys: list_P8527749157015355191n_assn] :
      ( ( R3 @ X @ Y )
     => ( ( listre1516905941988846596n_assn @ R3 @ Xs @ Ys )
       => ( listre1516905941988846596n_assn @ R3 @ ( cons_int @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_500_listrelp_OCons,axiom,
    ! [R3: produc6575502325842934193n_assn > nat > $o,X: produc6575502325842934193n_assn,Y: nat,Xs: list_P8527749157015355191n_assn,Ys: list_nat] :
      ( ( R3 @ X @ Y )
     => ( ( listre4659466805131468674sn_nat @ R3 @ Xs @ Ys )
       => ( listre4659466805131468674sn_nat @ R3 @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_501_listrelp_OCons,axiom,
    ! [R3: produc6575502325842934193n_assn > int > $o,X: produc6575502325842934193n_assn,Y: int,Xs: list_P8527749157015355191n_assn,Ys: list_int] :
      ( ( R3 @ X @ Y )
     => ( ( listre4656976334622418398sn_int @ R3 @ Xs @ Ys )
       => ( listre4656976334622418398sn_int @ R3 @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_502_listrelp_OCons,axiom,
    ! [R3: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( R3 @ X @ Y )
     => ( ( listre7738141641822031743n_assn @ R3 @ Xs @ Ys )
       => ( listre7738141641822031743n_assn @ R3 @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( cons_P2971678138204555879n_assn @ Y @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_503_listrelp_ONil,axiom,
    ! [R3: a > a > $o] : ( listrelp_a_a @ R3 @ nil_a @ nil_a ) ).

% listrelp.Nil
thf(fact_504_listrelp_ONil,axiom,
    ! [R3: a > b > $o] : ( listrelp_a_b @ R3 @ nil_a @ nil_b ) ).

% listrelp.Nil
thf(fact_505_listrelp_ONil,axiom,
    ! [R3: a > nat > $o] : ( listrelp_a_nat @ R3 @ nil_a @ nil_nat ) ).

% listrelp.Nil
thf(fact_506_listrelp_ONil,axiom,
    ! [R3: a > int > $o] : ( listrelp_a_int @ R3 @ nil_a @ nil_int ) ).

% listrelp.Nil
thf(fact_507_listrelp_ONil,axiom,
    ! [R3: b > a > $o] : ( listrelp_b_a @ R3 @ nil_b @ nil_a ) ).

% listrelp.Nil
thf(fact_508_listrelp_ONil,axiom,
    ! [R3: b > b > $o] : ( listrelp_b_b @ R3 @ nil_b @ nil_b ) ).

% listrelp.Nil
thf(fact_509_listrelp_ONil,axiom,
    ! [R3: b > nat > $o] : ( listrelp_b_nat @ R3 @ nil_b @ nil_nat ) ).

% listrelp.Nil
thf(fact_510_listrelp_ONil,axiom,
    ! [R3: b > int > $o] : ( listrelp_b_int @ R3 @ nil_b @ nil_int ) ).

% listrelp.Nil
thf(fact_511_listrelp_ONil,axiom,
    ! [R3: nat > a > $o] : ( listrelp_nat_a @ R3 @ nil_nat @ nil_a ) ).

% listrelp.Nil
thf(fact_512_listrelp_ONil,axiom,
    ! [R3: nat > b > $o] : ( listrelp_nat_b @ R3 @ nil_nat @ nil_b ) ).

% listrelp.Nil
thf(fact_513_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_zero_right
thf(fact_514_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_515_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_right
thf(fact_516_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_517_boolean__algebra_Oconj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_518_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
      = bot_bot_Product_unit ) ).

% boolean_algebra.conj_zero_left
thf(fact_519_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_520_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% boolean_algebra.conj_zero_left
thf(fact_521_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_522_boolean__algebra_Oconj__zero__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_523_inf__bot__right,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ bot_bot_Product_unit )
      = bot_bot_Product_unit ) ).

% inf_bot_right
thf(fact_524_inf__bot__right,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_525_inf__bot__right,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% inf_bot_right
thf(fact_526_inf__bot__right,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_527_inf__bot__right,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_528_inf__bot__left,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ bot_bot_Product_unit @ X )
      = bot_bot_Product_unit ) ).

% inf_bot_left
thf(fact_529_inf__bot__left,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_530_inf__bot__left,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% inf_bot_left
thf(fact_531_inf__bot__left,axiom,
    ! [X: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_532_inf__bot__left,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_533_inf__right__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Y )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_right_idem
thf(fact_534_inf__right__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_535_inf__right__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_536_inf__right__idem,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_right_idem
thf(fact_537_inf__right__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_right_idem
thf(fact_538_inf_Oright__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ B )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.right_idem
thf(fact_539_inf_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ B )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_540_inf_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_541_inf_Oright__idem,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ B )
      = ( inf_inf_Product_unit @ A @ B ) ) ).

% inf.right_idem
thf(fact_542_inf_Oright__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.right_idem
thf(fact_543_inf__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_left_idem
thf(fact_544_inf__left__idem,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_545_inf__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_546_inf__left__idem,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_left_idem
thf(fact_547_inf__left__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_left_idem
thf(fact_548_inf_Oleft__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( inf_inf_assn @ A @ ( inf_inf_assn @ A @ B ) )
      = ( inf_inf_assn @ A @ B ) ) ).

% inf.left_idem
thf(fact_549_inf_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( inf_inf_nat @ A @ ( inf_inf_nat @ A @ B ) )
      = ( inf_inf_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_550_inf_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
      = ( inf_inf_set_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_551_inf_Oleft__idem,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ A @ B ) )
      = ( inf_inf_Product_unit @ A @ B ) ) ).

% inf.left_idem
thf(fact_552_inf_Oleft__idem,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ A @ B ) )
      = ( inf_in2572325071724192079at_nat @ A @ B ) ) ).

% inf.left_idem
thf(fact_553_inf__idem,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ X )
      = X ) ).

% inf_idem
thf(fact_554_inf__idem,axiom,
    ! [X: nat] :
      ( ( inf_inf_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_555_inf__idem,axiom,
    ! [X: set_nat] :
      ( ( inf_inf_set_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_556_inf__idem,axiom,
    ! [X: product_unit] :
      ( ( inf_inf_Product_unit @ X @ X )
      = X ) ).

% inf_idem
thf(fact_557_inf__idem,axiom,
    ! [X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ X )
      = X ) ).

% inf_idem
thf(fact_558_inf_Oidem,axiom,
    ! [A: assn] :
      ( ( inf_inf_assn @ A @ A )
      = A ) ).

% inf.idem
thf(fact_559_inf_Oidem,axiom,
    ! [A: nat] :
      ( ( inf_inf_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_560_inf_Oidem,axiom,
    ! [A: set_nat] :
      ( ( inf_inf_set_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_561_inf_Oidem,axiom,
    ! [A: product_unit] :
      ( ( inf_inf_Product_unit @ A @ A )
      = A ) ).

% inf.idem
thf(fact_562_inf_Oidem,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A @ A )
      = A ) ).

% inf.idem
thf(fact_563_empty__Collect__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( collec939566748876313656_nat_o @ P ) )
      = ( ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_564_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X3: product_prod_nat_nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_565_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_566_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% empty_Collect_eq
thf(fact_567_empty__iff,axiom,
    ! [C2: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ C2 @ bot_bo7824918357723582233_nat_o ) ).

% empty_iff
thf(fact_568_empty__iff,axiom,
    ! [C2: $o] :
      ~ ( member_o @ C2 @ bot_bot_set_o ) ).

% empty_iff
thf(fact_569_empty__iff,axiom,
    ! [C2: nat] :
      ~ ( member_nat2 @ C2 @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_570_all__not__in__conv,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( member6576561426505652726_nat_o @ X3 @ A3 ) )
      = ( A3 = bot_bo7824918357723582233_nat_o ) ) ).

% all_not_in_conv
thf(fact_571_all__not__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ! [X3: $o] :
            ~ ( member_o @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_572_all__not__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ! [X3: nat] :
            ~ ( member_nat2 @ X3 @ A3 ) )
      = ( A3 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_573_Collect__empty__eq,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X3: produc3658429121746597890et_nat > $o] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_574_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_575_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_576_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ~ ( P @ X3 ) ) ) ).

% Collect_empty_eq
thf(fact_577_Int__emptyI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X2 @ A3 )
         => ~ ( member6576561426505652726_nat_o @ X2 @ B3 ) )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
        = bot_bo7824918357723582233_nat_o ) ) ).

% Int_emptyI
thf(fact_578_Int__emptyI,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A3 )
         => ~ ( member8440522571783428010at_nat @ X2 @ B3 ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_579_Int__emptyI,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A3 )
         => ~ ( member_o @ X2 @ B3 ) )
     => ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_580_Int__emptyI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A3 )
         => ~ ( member_nat2 @ X2 @ B3 ) )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_581_disjoint__iff,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
        = bot_bo7824918357723582233_nat_o )
      = ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ A3 )
           => ~ ( member6576561426505652726_nat_o @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_582_disjoint__iff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ~ ( member8440522571783428010at_nat @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_583_disjoint__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ~ ( member_o @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_584_disjoint__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ A3 )
           => ~ ( member_nat2 @ X3 @ B3 ) ) ) ) ).

% disjoint_iff
thf(fact_585_Int__empty__left,axiom,
    ! [B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B3 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_586_Int__empty__left,axiom,
    ! [B3: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B3 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_587_Int__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B3 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_588_Int__empty__right,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_589_Int__empty__right,axiom,
    ! [A3: set_o] :
      ( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_590_Int__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_591_disjoint__iff__not__equal,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ A3 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ B3 )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_592_disjoint__iff__not__equal,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ! [X3: $o] :
            ( ( member_o @ X3 @ A3 )
           => ! [Y3: $o] :
                ( ( member_o @ Y3 @ B3 )
               => ( X3 = ~ Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_593_disjoint__iff__not__equal,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ A3 )
           => ! [Y3: nat] :
                ( ( member_nat2 @ Y3 @ B3 )
               => ( X3 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_594_disjointI,axiom,
    ! [A: set_Pr4532377907799695533_nat_o,B: set_Pr4532377907799695533_nat_o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X2 @ A )
         => ~ ( member6576561426505652726_nat_o @ X2 @ B ) )
     => ( ( inf_in1906310914598751387_nat_o @ A @ B )
        = bot_bo7824918357723582233_nat_o ) ) ).

% disjointI
thf(fact_595_disjointI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ A )
         => ~ ( member8440522571783428010at_nat @ X2 @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjointI
thf(fact_596_disjointI,axiom,
    ! [A: set_o,B: set_o] :
      ( ! [X2: $o] :
          ( ( member_o @ X2 @ A )
         => ~ ( member_o @ X2 @ B ) )
     => ( ( inf_inf_set_o @ A @ B )
        = bot_bot_set_o ) ) ).

% disjointI
thf(fact_597_disjointI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A )
         => ~ ( member_nat2 @ X2 @ B ) )
     => ( ( inf_inf_set_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% disjointI
thf(fact_598_emptyE,axiom,
    ! [A: produc3658429121746597890et_nat > $o] :
      ~ ( member6576561426505652726_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ).

% emptyE
thf(fact_599_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_600_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat2 @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_601_equals0D,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o] :
      ( ( A3 = bot_bo7824918357723582233_nat_o )
     => ~ ( member6576561426505652726_nat_o @ A @ A3 ) ) ).

% equals0D
thf(fact_602_equals0D,axiom,
    ! [A3: set_o,A: $o] :
      ( ( A3 = bot_bot_set_o )
     => ~ ( member_o @ A @ A3 ) ) ).

% equals0D
thf(fact_603_equals0D,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( A3 = bot_bot_set_nat )
     => ~ ( member_nat2 @ A @ A3 ) ) ).

% equals0D
thf(fact_604_equals0I,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ! [Y2: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ Y2 @ A3 )
     => ( A3 = bot_bo7824918357723582233_nat_o ) ) ).

% equals0I
thf(fact_605_equals0I,axiom,
    ! [A3: set_o] :
      ( ! [Y2: $o] :
          ~ ( member_o @ Y2 @ A3 )
     => ( A3 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_606_equals0I,axiom,
    ! [A3: set_nat] :
      ( ! [Y2: nat] :
          ~ ( member_nat2 @ Y2 @ A3 )
     => ( A3 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_607_ex__in__conv,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( ? [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ A3 ) )
      = ( A3 != bot_bo7824918357723582233_nat_o ) ) ).

% ex_in_conv
thf(fact_608_ex__in__conv,axiom,
    ! [A3: set_o] :
      ( ( ? [X3: $o] : ( member_o @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_609_ex__in__conv,axiom,
    ! [A3: set_nat] :
      ( ( ? [X3: nat] : ( member_nat2 @ X3 @ A3 ) )
      = ( A3 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_610_bot__set__def,axiom,
    ( bot_bo7824918357723582233_nat_o
    = ( collec939566748876313656_nat_o @ bot_bo7963750851167320836at_o_o ) ) ).

% bot_set_def
thf(fact_611_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_612_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_613_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_614_inf__sup__aci_I4_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = ( inf_inf_assn @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_615_inf__sup__aci_I4_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = ( inf_inf_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_616_inf__sup__aci_I4_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = ( inf_inf_set_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_617_inf__sup__aci_I4_J,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = ( inf_inf_Product_unit @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_618_inf__sup__aci_I4_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = ( inf_in2572325071724192079at_nat @ X @ Y ) ) ).

% inf_sup_aci(4)
thf(fact_619_inf__sup__aci_I3_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_620_inf__sup__aci_I3_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_621_inf__sup__aci_I3_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_622_inf__sup__aci_I3_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_623_inf__sup__aci_I3_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_aci(3)
thf(fact_624_inf__sup__aci_I2_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_625_inf__sup__aci_I2_J,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_626_inf__sup__aci_I2_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_627_inf__sup__aci_I2_J,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
      = ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_628_inf__sup__aci_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(2)
thf(fact_629_inf__sup__aci_I1_J,axiom,
    ( inf_inf_assn
    = ( ^ [X3: assn,Y3: assn] : ( inf_inf_assn @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_630_inf__sup__aci_I1_J,axiom,
    ( inf_inf_nat
    = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_631_inf__sup__aci_I1_J,axiom,
    ( inf_inf_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_632_inf__sup__aci_I1_J,axiom,
    ( inf_inf_Product_unit
    = ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_633_inf__sup__aci_I1_J,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(1)
thf(fact_634_inf_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ A @ B ) @ C2 )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C2 ) ) ) ).

% inf.assoc
thf(fact_635_inf_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ A @ B ) @ C2 )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ).

% inf.assoc
thf(fact_636_inf_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).

% inf.assoc
thf(fact_637_inf_Oassoc,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ A @ B ) @ C2 )
      = ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) ) ) ).

% inf.assoc
thf(fact_638_inf_Oassoc,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ).

% inf.assoc
thf(fact_639_inf__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ ( inf_inf_assn @ X @ Y ) @ Z )
      = ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_640_inf__assoc,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
      = ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_641_inf__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
      = ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_642_inf__assoc,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Z )
      = ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_643_inf__assoc,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z )
      = ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ).

% inf_assoc
thf(fact_644_inf_Ocommute,axiom,
    ( inf_inf_assn
    = ( ^ [A2: assn,B2: assn] : ( inf_inf_assn @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_645_inf_Ocommute,axiom,
    ( inf_inf_nat
    = ( ^ [A2: nat,B2: nat] : ( inf_inf_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_646_inf_Ocommute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( inf_inf_set_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_647_inf_Ocommute,axiom,
    ( inf_inf_Product_unit
    = ( ^ [A2: product_unit,B2: product_unit] : ( inf_inf_Product_unit @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_648_inf_Ocommute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B2 @ A2 ) ) ) ).

% inf.commute
thf(fact_649_inf__commute,axiom,
    ( inf_inf_assn
    = ( ^ [X3: assn,Y3: assn] : ( inf_inf_assn @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_650_inf__commute,axiom,
    ( inf_inf_nat
    = ( ^ [X3: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_651_inf__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_652_inf__commute,axiom,
    ( inf_inf_Product_unit
    = ( ^ [X3: product_unit,Y3: product_unit] : ( inf_inf_Product_unit @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_653_inf__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ Y3 @ X3 ) ) ) ).

% inf_commute
thf(fact_654_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: assn,K: assn,A: assn,B: assn] :
      ( ( A3
        = ( inf_inf_assn @ K @ A ) )
     => ( ( inf_inf_assn @ A3 @ B )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_655_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( inf_inf_nat @ K @ A ) )
     => ( ( inf_inf_nat @ A3 @ B )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_656_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3
        = ( inf_inf_set_nat @ K @ A ) )
     => ( ( inf_inf_set_nat @ A3 @ B )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_657_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: product_unit,K: product_unit,A: product_unit,B: product_unit] :
      ( ( A3
        = ( inf_inf_Product_unit @ K @ A ) )
     => ( ( inf_inf_Product_unit @ A3 @ B )
        = ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_658_boolean__algebra__cancel_Oinf1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( inf_in2572325071724192079at_nat @ K @ A ) )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf1
thf(fact_659_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: assn,K: assn,B: assn,A: assn] :
      ( ( B3
        = ( inf_inf_assn @ K @ B ) )
     => ( ( inf_inf_assn @ A @ B3 )
        = ( inf_inf_assn @ K @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_660_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( inf_inf_nat @ K @ B ) )
     => ( ( inf_inf_nat @ A @ B3 )
        = ( inf_inf_nat @ K @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_661_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3
        = ( inf_inf_set_nat @ K @ B ) )
     => ( ( inf_inf_set_nat @ A @ B3 )
        = ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_662_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: product_unit,K: product_unit,B: product_unit,A: product_unit] :
      ( ( B3
        = ( inf_inf_Product_unit @ K @ B ) )
     => ( ( inf_inf_Product_unit @ A @ B3 )
        = ( inf_inf_Product_unit @ K @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_663_boolean__algebra__cancel_Oinf2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,K: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( inf_in2572325071724192079at_nat @ K @ B ) )
     => ( ( inf_in2572325071724192079at_nat @ A @ B3 )
        = ( inf_in2572325071724192079at_nat @ K @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.inf2
thf(fact_664_inf_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( inf_inf_assn @ B @ ( inf_inf_assn @ A @ C2 ) )
      = ( inf_inf_assn @ A @ ( inf_inf_assn @ B @ C2 ) ) ) ).

% inf.left_commute
thf(fact_665_inf_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( inf_inf_nat @ B @ ( inf_inf_nat @ A @ C2 ) )
      = ( inf_inf_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ).

% inf.left_commute
thf(fact_666_inf_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) )
      = ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).

% inf.left_commute
thf(fact_667_inf_Oleft__commute,axiom,
    ! [B: product_unit,A: product_unit,C2: product_unit] :
      ( ( inf_inf_Product_unit @ B @ ( inf_inf_Product_unit @ A @ C2 ) )
      = ( inf_inf_Product_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) ) ) ).

% inf.left_commute
thf(fact_668_inf_Oleft__commute,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ B @ ( inf_in2572325071724192079at_nat @ A @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ).

% inf.left_commute
thf(fact_669_inf__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ Y @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_670_inf__left__commute,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_671_inf__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_672_inf__left__commute,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ Y @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_673_inf__left__commute,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ Y @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_left_commute
thf(fact_674_FI__finalize,axiom,
    ! [M2: list_P8527749157015355191n_assn,P4: assn,Up: assn,Q3: assn,Uq: assn,F: assn] :
      ( ( fI_RESULT @ M2 @ ( times_times_assn @ P4 @ Up ) @ ( times_times_assn @ Q3 @ Uq ) @ F )
     => ( fi @ M2 @ P4 @ Q3 @ Up @ Uq @ F ) ) ).

% FI_finalize
thf(fact_675_Set_Ois__empty__def,axiom,
    ( is_empty_o
    = ( ^ [A5: set_o] : A5 = bot_bot_set_o ) ) ).

% Set.is_empty_def
thf(fact_676_Set_Ois__empty__def,axiom,
    ( is_empty_nat
    = ( ^ [A5: set_nat] : A5 = bot_bot_set_nat ) ) ).

% Set.is_empty_def
thf(fact_677_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_678_map__eq__map__tailrec,axiom,
    map_Pr8991440229025900053n_assn = map_ta2194206859901583947n_assn ).

% map_eq_map_tailrec
thf(fact_679_list__collect__set__map__simps_I3_J,axiom,
    ! [F: nat > set_nat,X: nat > nat,A: nat,L: list_nat] :
      ( ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ ( cons_nat @ A @ L ) ) )
      = ( sup_sup_set_nat @ ( F @ ( X @ A ) ) @ ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ L ) ) ) ) ).

% list_collect_set_map_simps(3)
thf(fact_680_list__collect__set__map__simps_I3_J,axiom,
    ! [F: assn > set_nat,X: produc6575502325842934193n_assn > assn,A: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
      ( ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ ( cons_P2971678138204555879n_assn @ A @ L ) ) )
      = ( sup_sup_set_nat @ ( F @ ( X @ A ) ) @ ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ L ) ) ) ) ).

% list_collect_set_map_simps(3)
thf(fact_681_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
thf(fact_682_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
    ( ( linord3142498349692569832_set_o @ bot_bot_set_o )
    = nil_o ) ).

% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_683_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
thf(fact_684_lexordp__eq_Ocases,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ( ord_lexordp_eq_assn @ A1 @ A22 )
     => ( ( A1 != nil_assn )
       => ( ! [X2: assn] :
              ( ? [Xs2: list_assn] :
                  ( A1
                  = ( cons_assn @ X2 @ Xs2 ) )
             => ! [Y2: assn] :
                  ( ? [Ys2: list_assn] :
                      ( A22
                      = ( cons_assn @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_assn @ X2 @ Y2 ) ) )
         => ~ ! [X2: assn,Y2: assn,Xs2: list_assn] :
                ( ( A1
                  = ( cons_assn @ X2 @ Xs2 ) )
               => ! [Ys2: list_assn] :
                    ( ( A22
                      = ( cons_assn @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_assn @ X2 @ Y2 )
                     => ( ~ ( ord_less_assn @ Y2 @ X2 )
                       => ~ ( ord_lexordp_eq_assn @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
thf(fact_685_lexordp__eq_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( ( ord_lexordp_eq_nat @ A1 @ A22 )
     => ( ( A1 != nil_nat )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] :
                  ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] :
                      ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_nat @ X2 @ Y2 ) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
               => ! [Ys2: list_nat] :
                    ( ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_nat @ X2 @ Y2 )
                     => ( ~ ( ord_less_nat @ Y2 @ X2 )
                       => ~ ( ord_lexordp_eq_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
thf(fact_686_lexordp__eq_Ocases,axiom,
    ! [A1: list_int,A22: list_int] :
      ( ( ord_lexordp_eq_int @ A1 @ A22 )
     => ( ( A1 != nil_int )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] :
                  ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] :
                      ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_int @ X2 @ Y2 ) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
               => ! [Ys2: list_int] :
                    ( ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_int @ X2 @ Y2 )
                     => ( ~ ( ord_less_int @ Y2 @ X2 )
                       => ~ ( ord_lexordp_eq_int @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp_eq.cases
thf(fact_687_lexordp__eq_Osimps,axiom,
    ( ord_lexordp_eq_assn
    = ( ^ [A12: list_assn,A23: list_assn] :
          ( ? [Ys3: list_assn] :
              ( ( A12 = nil_assn )
              & ( A23 = Ys3 ) )
          | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
              ( ( A12
                = ( cons_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_assn @ Y3 @ Ys3 ) )
              & ( ord_less_assn @ X3 @ Y3 ) )
          | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
              ( ( A12
                = ( cons_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_assn @ Y3 @ Ys3 ) )
              & ~ ( ord_less_assn @ X3 @ Y3 )
              & ~ ( ord_less_assn @ Y3 @ X3 )
              & ( ord_lexordp_eq_assn @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp_eq.simps
thf(fact_688_lexordp__eq_Osimps,axiom,
    ( ord_lexordp_eq_nat
    = ( ^ [A12: list_nat,A23: list_nat] :
          ( ? [Ys3: list_nat] :
              ( ( A12 = nil_nat )
              & ( A23 = Ys3 ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ( ord_less_nat @ X3 @ Y3 ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ~ ( ord_less_nat @ X3 @ Y3 )
              & ~ ( ord_less_nat @ Y3 @ X3 )
              & ( ord_lexordp_eq_nat @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp_eq.simps
thf(fact_689_lexordp__eq_Osimps,axiom,
    ( ord_lexordp_eq_int
    = ( ^ [A12: list_int,A23: list_int] :
          ( ? [Ys3: list_int] :
              ( ( A12 = nil_int )
              & ( A23 = Ys3 ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ( ord_less_int @ X3 @ Y3 ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ~ ( ord_less_int @ X3 @ Y3 )
              & ~ ( ord_less_int @ Y3 @ X3 )
              & ( ord_lexordp_eq_int @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp_eq.simps
thf(fact_690_bot__empty__eq,axiom,
    ( bot_bo7963750851167320836at_o_o
    = ( ^ [X3: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ X3 @ bot_bo7824918357723582233_nat_o ) ) ) ).

% bot_empty_eq
thf(fact_691_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X3: $o] : ( member_o @ X3 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_692_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X3: nat] : ( member_nat2 @ X3 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_693_Collect__empty__eq__bot,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ( collec939566748876313656_nat_o @ P )
        = bot_bo7824918357723582233_nat_o )
      = ( P = bot_bo7963750851167320836at_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_694_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_695_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_696_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_697_mult_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( ( syntax4682126007086162916at_nat @ ( times_times_nat @ X @ Y ) @ A )
     => ( ( times_times_nat @ A @ B )
        = ( times_times_nat @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_698_mult_Osafe__commute,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A )
     => ( ( times_times_assn @ A @ B )
        = ( times_times_assn @ B @ A ) ) ) ).

% mult.safe_commute
thf(fact_699_sup_Oidem,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ A @ A )
      = A ) ).

% sup.idem
thf(fact_700_sup_Oidem,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_701_sup__idem,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% sup_idem
thf(fact_702_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_703_sup_Oleft__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( sup_sup_assn @ A @ ( sup_sup_assn @ A @ B ) )
      = ( sup_sup_assn @ A @ B ) ) ).

% sup.left_idem
thf(fact_704_sup_Oleft__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_705_sup__left__idem,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% sup_left_idem
thf(fact_706_sup__left__idem,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_707_sup_Oright__idem,axiom,
    ! [A: assn,B: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A @ B ) @ B )
      = ( sup_sup_assn @ A @ B ) ) ).

% sup.right_idem
thf(fact_708_sup_Oright__idem,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ B )
      = ( sup_sup_set_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_709_Int__iff,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
      = ( ( member6576561426505652726_nat_o @ C2 @ A3 )
        & ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% Int_iff
thf(fact_710_Int__iff,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( member_nat2 @ C2 @ A3 )
        & ( member_nat2 @ C2 @ B3 ) ) ) ).

% Int_iff
thf(fact_711_Int__iff,axiom,
    ! [C2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( member8440522571783428010at_nat @ C2 @ A3 )
        & ( member8440522571783428010at_nat @ C2 @ B3 ) ) ) ).

% Int_iff
thf(fact_712_IntI,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ A3 )
     => ( ( member6576561426505652726_nat_o @ C2 @ B3 )
       => ( member6576561426505652726_nat_o @ C2 @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_713_IntI,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ A3 )
     => ( ( member_nat2 @ C2 @ B3 )
       => ( member_nat2 @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_714_IntI,axiom,
    ! [C2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ A3 )
     => ( ( member8440522571783428010at_nat @ C2 @ B3 )
       => ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% IntI
thf(fact_715_UnCI,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( ~ ( member6576561426505652726_nat_o @ C2 @ B3 )
       => ( member6576561426505652726_nat_o @ C2 @ A3 ) )
     => ( member6576561426505652726_nat_o @ C2 @ ( sup_su5209123915105501825_nat_o @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_716_UnCI,axiom,
    ! [C2: nat,B3: set_nat,A3: set_nat] :
      ( ( ~ ( member_nat2 @ C2 @ B3 )
       => ( member_nat2 @ C2 @ A3 ) )
     => ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_717_Un__iff,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( sup_su5209123915105501825_nat_o @ A3 @ B3 ) )
      = ( ( member6576561426505652726_nat_o @ C2 @ A3 )
        | ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% Un_iff
thf(fact_718_Un__iff,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( ( member_nat2 @ C2 @ A3 )
        | ( member_nat2 @ C2 @ B3 ) ) ) ).

% Un_iff
thf(fact_719_sup__bot_Oright__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ A @ bot_bot_assn )
      = A ) ).

% sup_bot.right_neutral
thf(fact_720_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_721_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_722_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ A @ B ) )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_723_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_724_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_725_sup__bot_Oleft__neutral,axiom,
    ! [A: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_726_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_727_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_728_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: assn,B: assn] :
      ( ( ( sup_sup_assn @ A @ B )
        = bot_bot_assn )
      = ( ( A = bot_bot_assn )
        & ( B = bot_bot_assn ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_729_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_730_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_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
thf(fact_731_sup__eq__bot__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( sup_sup_assn @ X @ Y )
        = bot_bot_assn )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% sup_eq_bot_iff
thf(fact_732_sup__eq__bot__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_733_sup__eq__bot__iff,axiom,
    ! [X: set_nat,Y: set_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
thf(fact_734_bot__eq__sup__iff,axiom,
    ! [X: assn,Y: assn] :
      ( ( bot_bot_assn
        = ( sup_sup_assn @ X @ Y ) )
      = ( ( X = bot_bot_assn )
        & ( Y = bot_bot_assn ) ) ) ).

% bot_eq_sup_iff
thf(fact_735_bot__eq__sup__iff,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X @ Y ) )
      = ( ( X = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_736_bot__eq__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X @ Y ) )
      = ( ( X = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_737_sup__bot__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% sup_bot_right
thf(fact_738_sup__bot__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% sup_bot_right
thf(fact_739_sup__bot__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% sup_bot_right
thf(fact_740_sup__bot__left,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% sup_bot_left
thf(fact_741_sup__bot__left,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X )
      = X ) ).

% sup_bot_left
thf(fact_742_sup__bot__left,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% sup_bot_left
thf(fact_743_sup__inf__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_744_sup__inf__absorb,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_745_sup__inf__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_746_sup__inf__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_747_sup__inf__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
      = X ) ).

% sup_inf_absorb
thf(fact_748_inf__sup__absorb,axiom,
    ! [X: nat,Y: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_749_inf__sup__absorb,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_750_inf__sup__absorb,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_751_inf__sup__absorb,axiom,
    ! [X: assn,Y: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_752_inf__sup__absorb,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = X ) ).

% inf_sup_absorb
thf(fact_753_Un__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( sup_sup_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ( A3 = bot_bot_set_o )
        & ( B3 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_754_Un__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ( A3 = bot_bot_set_nat )
        & ( B3 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_755_Un__Int__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_756_Un__Int__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
      = S ) ).

% Un_Int_eq(1)
thf(fact_757_Un__Int__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_758_Un__Int__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
      = T ) ).

% Un_Int_eq(2)
thf(fact_759_Un__Int__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_760_Un__Int__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
      = S ) ).

% Un_Int_eq(3)
thf(fact_761_Un__Int__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T @ ( sup_su6327502436637775413at_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_762_Un__Int__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
      = T ) ).

% Un_Int_eq(4)
thf(fact_763_Int__Un__eq_I1_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_764_Int__Un__eq_I1_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
      = S ) ).

% Int_Un_eq(1)
thf(fact_765_Int__Un__eq_I2_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_766_Int__Un__eq_I2_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
      = T ) ).

% Int_Un_eq(2)
thf(fact_767_Int__Un__eq_I3_J,axiom,
    ! [S: set_Pr1261947904930325089at_nat,T: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_768_Int__Un__eq_I3_J,axiom,
    ! [S: set_nat,T: set_nat] :
      ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
      = S ) ).

% Int_Un_eq(3)
thf(fact_769_Int__Un__eq_I4_J,axiom,
    ! [T: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T @ ( inf_in2572325071724192079at_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_770_Int__Un__eq_I4_J,axiom,
    ! [T: set_nat,S: set_nat] :
      ( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
      = T ) ).

% Int_Un_eq(4)
thf(fact_771_lexordp__eq__simps_I4_J,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] :
      ( ( ord_lexordp_eq_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) )
      = ( ( ord_less_assn @ X @ Y )
        | ( ~ ( ord_less_assn @ Y @ X )
          & ( ord_lexordp_eq_assn @ Xs @ Ys ) ) ) ) ).

% lexordp_eq_simps(4)
thf(fact_772_lexordp__eq__simps_I4_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ~ ( ord_less_nat @ Y @ X )
          & ( ord_lexordp_eq_nat @ Xs @ Ys ) ) ) ) ).

% lexordp_eq_simps(4)
thf(fact_773_lexordp__eq__simps_I4_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( ( ord_less_int @ X @ Y )
        | ( ~ ( ord_less_int @ Y @ X )
          & ( ord_lexordp_eq_int @ Xs @ Ys ) ) ) ) ).

% lexordp_eq_simps(4)
thf(fact_774_list__collect__set__simps_I3_J,axiom,
    ! [F: nat > set_nat,A: nat,L: list_nat] :
      ( ( list_c2452340269597857392at_nat @ F @ ( cons_nat @ A @ L ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( list_c2452340269597857392at_nat @ F @ L ) ) ) ).

% list_collect_set_simps(3)
thf(fact_775_list__collect__set__simps_I3_J,axiom,
    ! [F: int > set_nat,A: int,L: list_int] :
      ( ( list_c3451693981498911948nt_nat @ F @ ( cons_int @ A @ L ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( list_c3451693981498911948nt_nat @ F @ L ) ) ) ).

% list_collect_set_simps(3)
thf(fact_776_list__collect__set__simps_I3_J,axiom,
    ! [F: produc6575502325842934193n_assn > set_nat,A: produc6575502325842934193n_assn,L: list_P8527749157015355191n_assn] :
      ( ( list_c6061723043370948915sn_nat @ F @ ( cons_P2971678138204555879n_assn @ A @ L ) )
      = ( sup_sup_set_nat @ ( F @ A ) @ ( list_c6061723043370948915sn_nat @ F @ L ) ) ) ).

% list_collect_set_simps(3)
thf(fact_777_Int__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ B3 @ ( inf_inf_set_nat @ A3 @ C3 ) ) ) ).

% Int_left_commute
thf(fact_778_Int__left__commute,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ B3 @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) ) ) ).

% Int_left_commute
thf(fact_779_Un__Int__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) @ A3 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_780_Un__Int__distrib2,axiom,
    ! [B3: set_nat,C3: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ C3 ) @ A3 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ A3 ) @ ( sup_sup_set_nat @ C3 @ A3 ) ) ) ).

% Un_Int_distrib2
thf(fact_781_Int__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_782_Int__left__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Int_left_absorb
thf(fact_783_Int__Un__distrib2,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) @ A3 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B3 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_784_Int__Un__distrib2,axiom,
    ! [B3: set_nat,C3: set_nat,A3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B3 @ C3 ) @ A3 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B3 @ A3 ) @ ( inf_inf_set_nat @ C3 @ A3 ) ) ) ).

% Int_Un_distrib2
thf(fact_785_Un__Int__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_786_Un__Int__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ A3 @ C3 ) ) ) ).

% Un_Int_distrib
thf(fact_787_Int__Un__distrib,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_788_Int__Un__distrib,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ C3 ) ) ) ).

% Int_Un_distrib
thf(fact_789_Un__Int__crazy,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) ) @ ( sup_su6327502436637775413at_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_790_Un__Int__crazy,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) ) @ ( inf_inf_set_nat @ C3 @ A3 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ B3 @ C3 ) ) @ ( sup_sup_set_nat @ C3 @ A3 ) ) ) ).

% Un_Int_crazy
thf(fact_791_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] : ( inf_inf_set_nat @ B4 @ A5 ) ) ) ).

% Int_commute
thf(fact_792_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B4 @ A5 ) ) ) ).

% Int_commute
thf(fact_793_Int__absorb,axiom,
    ! [A3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_794_Int__absorb,axiom,
    ! [A3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
      = A3 ) ).

% Int_absorb
thf(fact_795_Int__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
      = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).

% Int_assoc
thf(fact_796_Int__assoc,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ).

% Int_assoc
thf(fact_797_IntD2,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ).

% IntD2
thf(fact_798_IntD2,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat2 @ C2 @ B3 ) ) ).

% IntD2
thf(fact_799_IntD2,axiom,
    ! [C2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C2 @ B3 ) ) ).

% IntD2
thf(fact_800_IntD1,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ( member6576561426505652726_nat_o @ C2 @ A3 ) ) ).

% IntD1
thf(fact_801_IntD1,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( member_nat2 @ C2 @ A3 ) ) ).

% IntD1
thf(fact_802_IntD1,axiom,
    ! [C2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( member8440522571783428010at_nat @ C2 @ A3 ) ) ).

% IntD1
thf(fact_803_IntE,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) )
     => ~ ( ( member6576561426505652726_nat_o @ C2 @ A3 )
         => ~ ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% IntE
thf(fact_804_IntE,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_nat2 @ C2 @ A3 )
         => ~ ( member_nat2 @ C2 @ B3 ) ) ) ).

% IntE
thf(fact_805_IntE,axiom,
    ! [C2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ~ ( ( member8440522571783428010at_nat @ C2 @ A3 )
         => ~ ( member8440522571783428010at_nat @ C2 @ B3 ) ) ) ).

% IntE
thf(fact_806_inf__sup__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_807_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
      = ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_808_inf__sup__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_809_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% inf_sup_aci(7)
thf(fact_810_inf__sup__aci_I6_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_811_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% inf_sup_aci(6)
thf(fact_812_inf__sup__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X3: assn,Y3: assn] : ( sup_sup_assn @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_813_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_814_sup_Oassoc,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ A @ B ) @ C2 )
      = ( sup_sup_assn @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_815_sup_Oassoc,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_816_sup__assoc,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_817_sup__assoc,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% sup_assoc
thf(fact_818_less__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ X @ A )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_819_less__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ X @ A )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI1
thf(fact_820_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_821_less__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_int @ X @ A )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI1
thf(fact_822_less__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ X @ B )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_823_less__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ X @ B )
     => ( ord_less_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% less_supI2
thf(fact_824_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X @ B )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_825_less__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_int @ X @ B )
     => ( ord_less_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% less_supI2
thf(fact_826_sup_Oabsorb3,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_827_sup_Oabsorb3,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_828_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_829_sup_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_830_sup_Oabsorb4,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_831_sup_Oabsorb4,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_832_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_833_sup_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_834_sup_Ocommute,axiom,
    ( sup_sup_assn
    = ( ^ [A2: assn,B2: assn] : ( sup_sup_assn @ B2 @ A2 ) ) ) ).

% sup.commute
thf(fact_835_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( sup_sup_set_nat @ B2 @ A2 ) ) ) ).

% sup.commute
thf(fact_836_sup__commute,axiom,
    ( sup_sup_assn
    = ( ^ [X3: assn,Y3: assn] : ( sup_sup_assn @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_837_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).

% sup_commute
thf(fact_838_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: assn,K: assn,A: assn,B: assn] :
      ( ( A3
        = ( sup_sup_assn @ K @ A ) )
     => ( ( sup_sup_assn @ A3 @ B )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_839_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_nat,K: set_nat,A: set_nat,B: set_nat] :
      ( ( A3
        = ( sup_sup_set_nat @ K @ A ) )
     => ( ( sup_sup_set_nat @ A3 @ B )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_840_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: assn,K: assn,B: assn,A: assn] :
      ( ( B3
        = ( sup_sup_assn @ K @ B ) )
     => ( ( sup_sup_assn @ A @ B3 )
        = ( sup_sup_assn @ K @ ( sup_sup_assn @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_841_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_nat,K: set_nat,B: set_nat,A: set_nat] :
      ( ( B3
        = ( sup_sup_set_nat @ K @ B ) )
     => ( ( sup_sup_set_nat @ A @ B3 )
        = ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_842_sup_Oleft__commute,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( sup_sup_assn @ B @ ( sup_sup_assn @ A @ C2 ) )
      = ( sup_sup_assn @ A @ ( sup_sup_assn @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_843_sup_Oleft__commute,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) )
      = ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_844_sup__left__commute,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_845_sup__left__commute,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_left_commute
thf(fact_846_sup_Ostrict__boundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_set_nat @ B @ A )
         => ~ ( ord_less_set_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_847_sup_Ostrict__boundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_assn @ ( sup_sup_assn @ B @ C2 ) @ A )
     => ~ ( ( ord_less_assn @ B @ A )
         => ~ ( ord_less_assn @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_848_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_849_sup_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ ( sup_sup_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_850_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( A2
            = ( sup_sup_set_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_851_sup_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( A2
            = ( sup_sup_assn @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_852_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( sup_sup_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_853_sup_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( sup_sup_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_854_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ C2 @ A )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_855_sup_Ostrict__coboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( ( ord_less_assn @ C2 @ A )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_856_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_857_sup_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ A )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_858_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ C2 @ B )
     => ( ord_less_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_859_sup_Ostrict__coboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( ( ord_less_assn @ C2 @ B )
     => ( ord_less_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_860_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_861_sup_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_int @ C2 @ B )
     => ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_862_UnE,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( sup_su5209123915105501825_nat_o @ A3 @ B3 ) )
     => ( ~ ( member6576561426505652726_nat_o @ C2 @ A3 )
       => ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% UnE
thf(fact_863_UnE,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ~ ( member_nat2 @ C2 @ A3 )
       => ( member_nat2 @ C2 @ B3 ) ) ) ).

% UnE
thf(fact_864_UnI1,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ A3 )
     => ( member6576561426505652726_nat_o @ C2 @ ( sup_su5209123915105501825_nat_o @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_865_UnI1,axiom,
    ! [C2: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ C2 @ A3 )
     => ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_866_UnI2,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ B3 )
     => ( member6576561426505652726_nat_o @ C2 @ ( sup_su5209123915105501825_nat_o @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_867_UnI2,axiom,
    ! [C2: nat,B3: set_nat,A3: set_nat] :
      ( ( member_nat2 @ C2 @ B3 )
     => ( member_nat2 @ C2 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_868_bex__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ A3 )
            & ( P @ X3 ) )
        | ? [X3: nat] :
            ( ( member_nat2 @ X3 @ B3 )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_869_ball__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ A3 )
           => ( P @ X3 ) )
        & ! [X3: nat] :
            ( ( member_nat2 @ X3 @ B3 )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_870_Un__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% Un_assoc
thf(fact_871_Un__absorb,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_872_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] : ( sup_sup_set_nat @ B4 @ A5 ) ) ) ).

% Un_commute
thf(fact_873_Un__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_874_Un__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_875_lt__ex,axiom,
    ! [X: int] :
    ? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).

% lt_ex
thf(fact_876_gt__ex,axiom,
    ! [X: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).

% gt_ex
thf(fact_877_gt__ex,axiom,
    ! [X: int] :
    ? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).

% gt_ex
thf(fact_878_less__imp__neq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_879_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_880_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_881_order_Oasym,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order.asym
thf(fact_882_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_883_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_884_ord__eq__less__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_885_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_886_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_887_ord__less__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_888_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_889_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_890_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X2 )
             => ( P @ Y4 ) )
         => ( P @ X2 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_891_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_892_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_893_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_894_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_895_dual__order_Oasym,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ~ ( ord_less_assn @ A @ B ) ) ).

% dual_order.asym
thf(fact_896_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_897_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_898_dual__order_Oirrefl,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ A ) ).

% dual_order.irrefl
thf(fact_899_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_900_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_901_exists__least__iff,axiom,
    ( ( ^ [P5: nat > $o] :
        ? [X4: nat] : ( P5 @ X4 ) )
    = ( ^ [P3: nat > $o] :
        ? [N3: nat] :
          ( ( P3 @ N3 )
          & ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ~ ( P3 @ M3 ) ) ) ) ) ).

% exists_least_iff
thf(fact_902_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B5: nat] :
          ( ( ord_less_nat @ A4 @ B5 )
         => ( P @ A4 @ B5 ) )
     => ( ! [A4: nat] : ( P @ A4 @ A4 )
       => ( ! [A4: nat,B5: nat] :
              ( ( P @ B5 @ A4 )
             => ( P @ A4 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_903_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B5: int] :
          ( ( ord_less_int @ A4 @ B5 )
         => ( P @ A4 @ B5 ) )
     => ( ! [A4: int] : ( P @ A4 @ A4 )
       => ( ! [A4: int,B5: int] :
              ( ( P @ B5 @ A4 )
             => ( P @ A4 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_904_order_Ostrict__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_905_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_906_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_907_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_908_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_909_dual__order_Ostrict__trans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_910_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_911_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_912_order_Ostrict__implies__not__eq,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_913_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_914_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_915_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_916_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_917_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_918_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_919_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_920_order__less__asym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_asym
thf(fact_921_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_922_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_923_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_924_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_925_order__less__asym_H,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ~ ( ord_less_assn @ B @ A ) ) ).

% order_less_asym'
thf(fact_926_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_927_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_928_order__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_929_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_930_order__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_trans
thf(fact_931_ord__eq__less__subst,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_932_ord__eq__less__subst,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_933_ord__eq__less__subst,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_934_ord__eq__less__subst,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_935_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_936_ord__eq__less__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_937_ord__eq__less__subst,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_938_ord__eq__less__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_939_ord__eq__less__subst,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_940_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_941_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_942_ord__less__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_943_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_944_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_945_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_946_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_947_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_948_ord__less__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_949_order__less__irrefl,axiom,
    ! [X: assn] :
      ~ ( ord_less_assn @ X @ X ) ).

% order_less_irrefl
thf(fact_950_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_951_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_952_order__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_953_order__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_954_order__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_955_order__less__subst1,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_956_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_957_order__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_958_order__less__subst1,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_959_order__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_960_order__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_961_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_962_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_963_order__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_964_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_965_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_966_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_967_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_968_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_969_order__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_970_order__less__not__sym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_971_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_972_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_973_order__less__imp__triv,axiom,
    ! [X: assn,Y: assn,P: $o] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_974_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_975_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P ) ) ).

% order_less_imp_triv
thf(fact_976_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_977_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_978_order__less__imp__not__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_979_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_980_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_981_order__less__imp__not__eq2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_982_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_983_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_984_order__less__imp__not__less,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ~ ( ord_less_assn @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_985_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_986_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_987_syntax__fo__nomatch__def,axiom,
    ( syntax7398250324933576852n_assn
    = ( ^ [Pat: assn,Obj: assn] : $true ) ) ).

% syntax_fo_nomatch_def
thf(fact_988_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_989_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_o] :
      ( ( sup_sup_set_o @ X @ bot_bot_set_o )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_990_boolean__algebra_Odisj__zero__right,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% boolean_algebra.disj_zero_right
thf(fact_991_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_992_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_993_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_994_boolean__algebra_Odisj__conj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.disj_conj_distrib2
thf(fact_995_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_996_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_997_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_998_boolean__algebra_Oconj__disj__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% boolean_algebra.conj_disj_distrib2
thf(fact_999_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1000_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1001_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1002_boolean__algebra_Odisj__conj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.disj_conj_distrib
thf(fact_1003_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1004_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1005_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1006_boolean__algebra_Oconj__disj__distrib,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% boolean_algebra.conj_disj_distrib
thf(fact_1007_sup__inf__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
      = ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1008_sup__inf__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ Z ) @ X )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ X ) @ ( sup_sup_Product_unit @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1009_sup__inf__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ Z ) @ X )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ X ) @ ( sup_su6327502436637775413at_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1010_sup__inf__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( sup_sup_assn @ ( inf_inf_assn @ Y @ Z ) @ X )
      = ( inf_inf_assn @ ( sup_sup_assn @ Y @ X ) @ ( sup_sup_assn @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1011_sup__inf__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).

% sup_inf_distrib2
thf(fact_1012_sup__inf__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1013_sup__inf__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1014_sup__inf__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1015_sup__inf__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1016_sup__inf__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% sup_inf_distrib1
thf(fact_1017_inf__sup__distrib2,axiom,
    ! [Y: nat,Z: nat,X: nat] :
      ( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
      = ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1018_inf__sup__distrib2,axiom,
    ! [Y: product_unit,Z: product_unit,X: product_unit] :
      ( ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ Y @ Z ) @ X )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ Y @ X ) @ ( inf_inf_Product_unit @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1019_inf__sup__distrib2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ Y @ Z ) @ X )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ Y @ X ) @ ( inf_in2572325071724192079at_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1020_inf__sup__distrib2,axiom,
    ! [Y: assn,Z: assn,X: assn] :
      ( ( inf_inf_assn @ ( sup_sup_assn @ Y @ Z ) @ X )
      = ( sup_sup_assn @ ( inf_inf_assn @ Y @ X ) @ ( inf_inf_assn @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1021_inf__sup__distrib2,axiom,
    ! [Y: set_nat,Z: set_nat,X: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).

% inf_sup_distrib2
thf(fact_1022_inf__sup__distrib1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
      = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1023_inf__sup__distrib1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
      = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1024_inf__sup__distrib1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1025_inf__sup__distrib1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1026_inf__sup__distrib1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).

% inf_sup_distrib1
thf(fact_1027_distrib__imp2,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z3: nat] :
          ( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y2 @ Z3 ) )
          = ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( sup_sup_nat @ X2 @ Z3 ) ) )
     => ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
        = ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1028_distrib__imp2,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] :
          ( ( sup_sup_Product_unit @ X2 @ ( inf_inf_Product_unit @ Y2 @ Z3 ) )
          = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X2 @ Y2 ) @ ( sup_sup_Product_unit @ X2 @ Z3 ) ) )
     => ( ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) )
        = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1029_distrib__imp2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
          ( ( sup_su6327502436637775413at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y2 @ Z3 ) )
          = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ Y2 ) @ ( sup_su6327502436637775413at_nat @ X2 @ Z3 ) ) )
     => ( ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) )
        = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1030_distrib__imp2,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] :
          ( ( sup_sup_assn @ X2 @ ( inf_inf_assn @ Y2 @ Z3 ) )
          = ( inf_inf_assn @ ( sup_sup_assn @ X2 @ Y2 ) @ ( sup_sup_assn @ X2 @ Z3 ) ) )
     => ( ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
        = ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1031_distrib__imp2,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
          ( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y2 @ Z3 ) )
          = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( sup_sup_set_nat @ X2 @ Z3 ) ) )
     => ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
        = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp2
thf(fact_1032_distrib__imp1,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ! [X2: nat,Y2: nat,Z3: nat] :
          ( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y2 @ Z3 ) )
          = ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( inf_inf_nat @ X2 @ Z3 ) ) )
     => ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
        = ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1033_distrib__imp1,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] :
          ( ( inf_inf_Product_unit @ X2 @ ( sup_sup_Product_unit @ Y2 @ Z3 ) )
          = ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y2 ) @ ( inf_inf_Product_unit @ X2 @ Z3 ) ) )
     => ( ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
        = ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1034_distrib__imp1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y2 @ Z3 ) )
          = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Z3 ) ) )
     => ( ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
        = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1035_distrib__imp1,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ! [X2: assn,Y2: assn,Z3: assn] :
          ( ( inf_inf_assn @ X2 @ ( sup_sup_assn @ Y2 @ Z3 ) )
          = ( sup_sup_assn @ ( inf_inf_assn @ X2 @ Y2 ) @ ( inf_inf_assn @ X2 @ Z3 ) ) )
     => ( ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
        = ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1036_distrib__imp1,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
          ( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y2 @ Z3 ) )
          = ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( inf_inf_set_nat @ X2 @ Z3 ) ) )
     => ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).

% distrib_imp1
thf(fact_1037_Un__empty__right,axiom,
    ! [A3: set_o] :
      ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Un_empty_right
thf(fact_1038_Un__empty__right,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Un_empty_right
thf(fact_1039_Un__empty__left,axiom,
    ! [B3: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1040_Un__empty__left,axiom,
    ! [B3: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B3 )
      = B3 ) ).

% Un_empty_left
thf(fact_1041_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1042_bot_Onot__eq__extremum,axiom,
    ! [A: set_nat] :
      ( ( A != bot_bot_set_nat )
      = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1043_bot_Onot__eq__extremum,axiom,
    ! [A: assn] :
      ( ( A != bot_bot_assn )
      = ( ord_less_assn @ bot_bot_assn @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1044_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1045_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1046_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1047_bot_Oextremum__strict,axiom,
    ! [A: assn] :
      ~ ( ord_less_assn @ A @ bot_bot_assn ) ).

% bot.extremum_strict
thf(fact_1048_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1049_not__psubset__empty,axiom,
    ! [A3: set_o] :
      ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1050_not__psubset__empty,axiom,
    ! [A3: set_nat] :
      ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1051_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1052_inf_Ostrict__coboundedI2,axiom,
    ! [B: product_unit,C2: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ C2 )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1053_inf_Ostrict__coboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ C2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1054_inf_Ostrict__coboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_assn @ B @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1055_inf_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ B @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1056_inf_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ B @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI2
thf(fact_1057_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ C2 )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1058_inf_Ostrict__coboundedI1,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ C2 )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1059_inf_Ostrict__coboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ C2 )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1060_inf_Ostrict__coboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( ord_less_assn @ A @ C2 )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1061_inf_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ A @ C2 )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1062_inf_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ C2 )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.strict_coboundedI1
thf(fact_1063_inf_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( A2
            = ( inf_inf_set_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1064_inf_Ostrict__order__iff,axiom,
    ( ord_le361264281704409273t_unit
    = ( ^ [A2: product_unit,B2: product_unit] :
          ( ( A2
            = ( inf_inf_Product_unit @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1065_inf_Ostrict__order__iff,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( A2
            = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1066_inf_Ostrict__order__iff,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( A2
            = ( inf_inf_assn @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1067_inf_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( A2
            = ( inf_inf_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1068_inf_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( A2
            = ( inf_inf_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% inf.strict_order_iff
thf(fact_1069_inf_Ostrict__boundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
     => ~ ( ( ord_less_set_nat @ A @ B )
         => ~ ( ord_less_set_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1070_inf_Ostrict__boundedE,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) )
     => ~ ( ( ord_le361264281704409273t_unit @ A @ B )
         => ~ ( ord_le361264281704409273t_unit @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1071_inf_Ostrict__boundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
     => ~ ( ( ord_le7866589430770878221at_nat @ A @ B )
         => ~ ( ord_le7866589430770878221at_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1072_inf_Ostrict__boundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
     => ~ ( ( ord_less_assn @ A @ B )
         => ~ ( ord_less_assn @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1073_inf_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1074_inf_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( inf_inf_int @ B @ C2 ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C2 ) ) ) ).

% inf.strict_boundedE
thf(fact_1075_inf_Oabsorb4,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1076_inf_Oabsorb4,axiom,
    ! [B: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ A )
     => ( ( inf_inf_Product_unit @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1077_inf_Oabsorb4,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1078_inf_Oabsorb4,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1079_inf_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1080_inf_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb4
thf(fact_1081_inf_Oabsorb3,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1082_inf_Oabsorb3,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ B )
     => ( ( inf_inf_Product_unit @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1083_inf_Oabsorb3,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1084_inf_Oabsorb3,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1085_inf_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1086_inf_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb3
thf(fact_1087_less__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1088_less__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( ( ord_le361264281704409273t_unit @ B @ X )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1089_less__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ B @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1090_less__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_assn @ B @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1091_less__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_nat @ B @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1092_less__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_int @ B @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI2
thf(fact_1093_less__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ X )
     => ( ord_less_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1094_less__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( ( ord_le361264281704409273t_unit @ A @ X )
     => ( ord_le361264281704409273t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1095_less__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A @ X )
     => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1096_less__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_assn @ A @ X )
     => ( ord_less_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1097_less__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_nat @ A @ X )
     => ( ord_less_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1098_less__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_int @ A @ X )
     => ( ord_less_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% less_infI1
thf(fact_1099_FI__p__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Ps: assn,Qs: assn,Q3: assn,P4: assn,Up: assn,Uq: assn,F: assn] :
      ( ( fi @ M2 @ Ps @ ( times_times_assn @ Qs @ Q3 ) @ ( times_times_assn @ P4 @ Up ) @ Uq @ F )
     => ( fi @ M2 @ ( times_times_assn @ Ps @ P4 ) @ ( times_times_assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ).

% FI_p_nomatch
thf(fact_1100_lexordp__eq_OCons__eq,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ~ ( ord_less_assn @ Y @ X )
       => ( ( ord_lexordp_eq_assn @ Xs @ Ys )
         => ( ord_lexordp_eq_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) ) ) ) ) ).

% lexordp_eq.Cons_eq
thf(fact_1101_lexordp__eq_OCons__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ~ ( ord_less_nat @ Y @ X )
       => ( ( ord_lexordp_eq_nat @ Xs @ Ys )
         => ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).

% lexordp_eq.Cons_eq
thf(fact_1102_lexordp__eq_OCons__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ~ ( ord_less_int @ Y @ X )
       => ( ( ord_lexordp_eq_int @ Xs @ Ys )
         => ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).

% lexordp_eq.Cons_eq
thf(fact_1103_lexordp__eq_OCons,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ord_lexordp_eq_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) ) ) ).

% lexordp_eq.Cons
thf(fact_1104_lexordp__eq_OCons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_lexordp_eq_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).

% lexordp_eq.Cons
thf(fact_1105_lexordp__eq_OCons,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_lexordp_eq_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).

% lexordp_eq.Cons
thf(fact_1106_assn__aci_I11_J,axiom,
    ! [X: assn,Y: assn,A: assn,B: assn] :
      ( ( syntax7398250324933576852n_assn @ ( times_times_assn @ X @ Y ) @ A )
     => ( ( times_times_assn @ A @ B )
        = ( times_times_assn @ B @ A ) ) ) ).

% assn_aci(11)
thf(fact_1107_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1108_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_1109_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1110_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_1111_FI__q__nomatch,axiom,
    ! [M2: list_P8527749157015355191n_assn,Up: assn,Qs: assn,Q3: assn,Uq: assn,F: assn] :
      ( ( fi @ M2 @ ( times_times_assn @ sln @ Up ) @ Qs @ sln @ ( times_times_assn @ Q3 @ Uq ) @ F )
     => ( fi @ M2 @ sln @ ( times_times_assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ).

% FI_q_nomatch
thf(fact_1112_merge_Oelims,axiom,
    ! [X: list_nat,Xa: list_nat,Y: list_nat] :
      ( ( ( merge_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != Xa ) )
       => ( ! [V2: nat,Va2: list_nat] :
              ( ( X
                = ( cons_nat @ V2 @ Va2 ) )
             => ( ( Xa = nil_nat )
               => ( Y
                 != ( cons_nat @ V2 @ Va2 ) ) ) )
         => ~ ! [X1: nat,L1: list_nat] :
                ( ( X
                  = ( cons_nat @ X1 @ L1 ) )
               => ! [X23: nat,L22: list_nat] :
                    ( ( Xa
                      = ( cons_nat @ X23 @ L22 ) )
                   => ~ ( ( ( ord_less_nat @ X1 @ X23 )
                         => ( Y
                            = ( cons_nat @ X1 @ ( merge_nat @ L1 @ ( cons_nat @ X23 @ L22 ) ) ) ) )
                        & ( ~ ( ord_less_nat @ X1 @ X23 )
                         => ( ( ( X1 = X23 )
                             => ( Y
                                = ( cons_nat @ X1 @ ( merge_nat @ L1 @ L22 ) ) ) )
                            & ( ( X1 != X23 )
                             => ( Y
                                = ( cons_nat @ X23 @ ( merge_nat @ ( cons_nat @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_1113_merge_Oelims,axiom,
    ! [X: list_int,Xa: list_int,Y: list_int] :
      ( ( ( merge_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_int )
         => ( Y != Xa ) )
       => ( ! [V2: int,Va2: list_int] :
              ( ( X
                = ( cons_int @ V2 @ Va2 ) )
             => ( ( Xa = nil_int )
               => ( Y
                 != ( cons_int @ V2 @ Va2 ) ) ) )
         => ~ ! [X1: int,L1: list_int] :
                ( ( X
                  = ( cons_int @ X1 @ L1 ) )
               => ! [X23: int,L22: list_int] :
                    ( ( Xa
                      = ( cons_int @ X23 @ L22 ) )
                   => ~ ( ( ( ord_less_int @ X1 @ X23 )
                         => ( Y
                            = ( cons_int @ X1 @ ( merge_int @ L1 @ ( cons_int @ X23 @ L22 ) ) ) ) )
                        & ( ~ ( ord_less_int @ X1 @ X23 )
                         => ( ( ( X1 = X23 )
                             => ( Y
                                = ( cons_int @ X1 @ ( merge_int @ L1 @ L22 ) ) ) )
                            & ( ( X1 != X23 )
                             => ( Y
                                = ( cons_int @ X23 @ ( merge_int @ ( cons_int @ X1 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).

% merge.elims
thf(fact_1114_list__collect__set__map__simps_I4_J,axiom,
    ! [F: nat > set_nat,X: nat > nat,L: list_nat,L3: list_nat] :
      ( ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ ( append_nat @ L @ L3 ) ) )
      = ( sup_sup_set_nat @ ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ L ) ) @ ( list_c2452340269597857392at_nat @ F @ ( map_nat_nat @ X @ L3 ) ) ) ) ).

% list_collect_set_map_simps(4)
thf(fact_1115_list__collect__set__map__simps_I4_J,axiom,
    ! [F: assn > set_nat,X: produc6575502325842934193n_assn > assn,L: list_P8527749157015355191n_assn,L3: list_P8527749157015355191n_assn] :
      ( ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ ( append282499809098378956n_assn @ L @ L3 ) ) )
      = ( sup_sup_set_nat @ ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ L ) ) @ ( list_c1844713377658005960sn_nat @ F @ ( map_Pr8991440229025900053n_assn @ X @ L3 ) ) ) ) ).

% list_collect_set_map_simps(4)
thf(fact_1116_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( ( linord2612477271533052124et_int @ A3 )
          = nil_int )
        = ( A3 = bot_bot_set_int ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1117_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( linord3142498349692569832_set_o @ A3 )
          = nil_o )
        = ( A3 = bot_bot_set_o ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1118_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( linord2614967742042102400et_nat @ A3 )
          = nil_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_1119_lexordp__induct,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ord_lexordp_nat @ Xs @ Ys )
     => ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys2 ) )
       => ( ! [X2: nat,Xs2: list_nat,Y2: nat,Ys2: list_nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ Y2 @ Ys2 ) ) )
         => ( ! [X2: nat,Xs2: list_nat,Ys2: list_nat] :
                ( ( ord_lexordp_nat @ Xs2 @ Ys2 )
               => ( ( P @ Xs2 @ Ys2 )
                 => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_nat @ X2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% lexordp_induct
thf(fact_1120_lexordp__induct,axiom,
    ! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
      ( ( ord_lexordp_int @ Xs @ Ys )
     => ( ! [Y2: int,Ys2: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
       => ( ! [X2: int,Xs2: list_int,Y2: int,Ys2: list_int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
         => ( ! [X2: int,Xs2: list_int,Ys2: list_int] :
                ( ( ord_lexordp_int @ Xs2 @ Ys2 )
               => ( ( P @ Xs2 @ Ys2 )
                 => ( P @ ( cons_int @ X2 @ Xs2 ) @ ( cons_int @ X2 @ Ys2 ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% lexordp_induct
thf(fact_1121_lexordp__cases,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ord_lexordp_nat @ Xs @ Ys )
     => ( ( ( Xs = nil_nat )
         => ! [Y2: nat,Ys4: list_nat] :
              ( Ys
             != ( cons_nat @ Y2 @ Ys4 ) ) )
       => ( ! [X2: nat] :
              ( ? [Xs4: list_nat] :
                  ( Xs
                  = ( cons_nat @ X2 @ Xs4 ) )
             => ! [Y2: nat] :
                  ( ? [Ys4: list_nat] :
                      ( Ys
                      = ( cons_nat @ Y2 @ Ys4 ) )
                 => ~ ( ord_less_nat @ X2 @ Y2 ) ) )
         => ~ ! [X2: nat,Xs4: list_nat] :
                ( ( Xs
                  = ( cons_nat @ X2 @ Xs4 ) )
               => ! [Ys4: list_nat] :
                    ( ( Ys
                      = ( cons_nat @ X2 @ Ys4 ) )
                   => ~ ( ord_lexordp_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% lexordp_cases
thf(fact_1122_lexordp__cases,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ord_lexordp_int @ Xs @ Ys )
     => ( ( ( Xs = nil_int )
         => ! [Y2: int,Ys4: list_int] :
              ( Ys
             != ( cons_int @ Y2 @ Ys4 ) ) )
       => ( ! [X2: int] :
              ( ? [Xs4: list_int] :
                  ( Xs
                  = ( cons_int @ X2 @ Xs4 ) )
             => ! [Y2: int] :
                  ( ? [Ys4: list_int] :
                      ( Ys
                      = ( cons_int @ Y2 @ Ys4 ) )
                 => ~ ( ord_less_int @ X2 @ Y2 ) ) )
         => ~ ! [X2: int,Xs4: list_int] :
                ( ( Xs
                  = ( cons_int @ X2 @ Xs4 ) )
               => ! [Ys4: list_int] :
                    ( ( Ys
                      = ( cons_int @ X2 @ Ys4 ) )
                   => ~ ( ord_lexordp_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% lexordp_cases
thf(fact_1123_lexordp_Osimps,axiom,
    ( ord_lexordp_assn
    = ( ^ [A12: list_assn,A23: list_assn] :
          ( ? [Y3: assn,Ys3: list_assn] :
              ( ( A12 = nil_assn )
              & ( A23
                = ( cons_assn @ Y3 @ Ys3 ) ) )
          | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
              ( ( A12
                = ( cons_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_assn @ Y3 @ Ys3 ) )
              & ( ord_less_assn @ X3 @ Y3 ) )
          | ? [X3: assn,Y3: assn,Xs3: list_assn,Ys3: list_assn] :
              ( ( A12
                = ( cons_assn @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_assn @ Y3 @ Ys3 ) )
              & ~ ( ord_less_assn @ X3 @ Y3 )
              & ~ ( ord_less_assn @ Y3 @ X3 )
              & ( ord_lexordp_assn @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp.simps
thf(fact_1124_lexordp_Osimps,axiom,
    ( ord_lexordp_nat
    = ( ^ [A12: list_nat,A23: list_nat] :
          ( ? [Y3: nat,Ys3: list_nat] :
              ( ( A12 = nil_nat )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ( ord_less_nat @ X3 @ Y3 ) )
          | ? [X3: nat,Y3: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y3 @ Ys3 ) )
              & ~ ( ord_less_nat @ X3 @ Y3 )
              & ~ ( ord_less_nat @ Y3 @ X3 )
              & ( ord_lexordp_nat @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp.simps
thf(fact_1125_lexordp_Osimps,axiom,
    ( ord_lexordp_int
    = ( ^ [A12: list_int,A23: list_int] :
          ( ? [Y3: int,Ys3: list_int] :
              ( ( A12 = nil_int )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ( ord_less_int @ X3 @ Y3 ) )
          | ? [X3: int,Y3: int,Xs3: list_int,Ys3: list_int] :
              ( ( A12
                = ( cons_int @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_int @ Y3 @ Ys3 ) )
              & ~ ( ord_less_int @ X3 @ Y3 )
              & ~ ( ord_less_int @ Y3 @ X3 )
              & ( ord_lexordp_int @ Xs3 @ Ys3 ) ) ) ) ) ).

% lexordp.simps
thf(fact_1126_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs3 ) )
      = ( Ys = Zs3 ) ) ).

% same_append_eq
thf(fact_1127_same__append__eq,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = ( append_int @ Xs @ Zs3 ) )
      = ( Ys = Zs3 ) ) ).

% same_append_eq
thf(fact_1128_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs3 @ Xs ) )
      = ( Ys = Zs3 ) ) ).

% append_same_eq
thf(fact_1129_append__same__eq,axiom,
    ! [Ys: list_int,Xs: list_int,Zs3: list_int] :
      ( ( ( append_int @ Ys @ Xs )
        = ( append_int @ Zs3 @ Xs ) )
      = ( Ys = Zs3 ) ) ).

% append_same_eq
thf(fact_1130_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs3 )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs3 ) ) ) ).

% append_assoc
thf(fact_1131_append__assoc,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( append_int @ ( append_int @ Xs @ Ys ) @ Zs3 )
      = ( append_int @ Xs @ ( append_int @ Ys @ Zs3 ) ) ) ).

% append_assoc
thf(fact_1132_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_1133_append_Oassoc,axiom,
    ! [A: list_int,B: list_int,C2: list_int] :
      ( ( append_int @ ( append_int @ A @ B ) @ C2 )
      = ( append_int @ A @ ( append_int @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_1134_merge__pure__or,axiom,
    ! [A: $o,B: $o] :
      ( ( sup_sup_assn @ ( pure_assn @ A ) @ ( pure_assn @ B ) )
      = ( pure_assn
        @ ( A
          | B ) ) ) ).

% merge_pure_or
thf(fact_1135_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_1136_append__is__Nil__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( append_b @ Xs @ Ys )
        = nil_b )
      = ( ( Xs = nil_b )
        & ( Ys = nil_b ) ) ) ).

% append_is_Nil_conv
thf(fact_1137_append__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Xs @ Ys )
        = nil_Pr5671120429643327159n_assn )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% append_is_Nil_conv
thf(fact_1138_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_1139_append__is__Nil__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = nil_int )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% append_is_Nil_conv
thf(fact_1140_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_1141_Nil__is__append__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( nil_b
        = ( append_b @ Xs @ Ys ) )
      = ( ( Xs = nil_b )
        & ( Ys = nil_b ) ) ) ).

% Nil_is_append_conv
thf(fact_1142_Nil__is__append__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( append282499809098378956n_assn @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr5671120429643327159n_assn )
        & ( Ys = nil_Pr5671120429643327159n_assn ) ) ) ).

% Nil_is_append_conv
thf(fact_1143_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_1144_Nil__is__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( nil_int
        = ( append_int @ Xs @ Ys ) )
      = ( ( Xs = nil_int )
        & ( Ys = nil_int ) ) ) ).

% Nil_is_append_conv
thf(fact_1145_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_1146_self__append__conv2,axiom,
    ! [Y: list_b,Xs: list_b] :
      ( ( Y
        = ( append_b @ Xs @ Y ) )
      = ( Xs = nil_b ) ) ).

% self_append_conv2
thf(fact_1147_self__append__conv2,axiom,
    ! [Y: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Y
        = ( append282499809098378956n_assn @ Xs @ Y ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% self_append_conv2
thf(fact_1148_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_1149_self__append__conv2,axiom,
    ! [Y: list_int,Xs: list_int] :
      ( ( Y
        = ( append_int @ Xs @ Y ) )
      = ( Xs = nil_int ) ) ).

% self_append_conv2
thf(fact_1150_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_1151_append__self__conv2,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( append_b @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_b ) ) ).

% append_self_conv2
thf(fact_1152_append__self__conv2,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% append_self_conv2
thf(fact_1153_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_1154_append__self__conv2,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_int ) ) ).

% append_self_conv2
thf(fact_1155_self__append__conv,axiom,
    ! [Y: list_a,Ys: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_1156_self__append__conv,axiom,
    ! [Y: list_b,Ys: list_b] :
      ( ( Y
        = ( append_b @ Y @ Ys ) )
      = ( Ys = nil_b ) ) ).

% self_append_conv
thf(fact_1157_self__append__conv,axiom,
    ! [Y: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( Y
        = ( append282499809098378956n_assn @ Y @ Ys ) )
      = ( Ys = nil_Pr5671120429643327159n_assn ) ) ).

% self_append_conv
thf(fact_1158_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_1159_self__append__conv,axiom,
    ! [Y: list_int,Ys: list_int] :
      ( ( Y
        = ( append_int @ Y @ Ys ) )
      = ( Ys = nil_int ) ) ).

% self_append_conv
thf(fact_1160_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_1161_append__self__conv,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( ( append_b @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_b ) ) ).

% append_self_conv
thf(fact_1162_append__self__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr5671120429643327159n_assn ) ) ).

% append_self_conv
thf(fact_1163_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_1164_append__self__conv,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_int ) ) ).

% append_self_conv
thf(fact_1165_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_1166_append__Nil2,axiom,
    ! [Xs: list_b] :
      ( ( append_b @ Xs @ nil_b )
      = Xs ) ).

% append_Nil2
thf(fact_1167_append__Nil2,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( append282499809098378956n_assn @ Xs @ nil_Pr5671120429643327159n_assn )
      = Xs ) ).

% append_Nil2
thf(fact_1168_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_1169_append__Nil2,axiom,
    ! [Xs: list_int] :
      ( ( append_int @ Xs @ nil_int )
      = Xs ) ).

% append_Nil2
thf(fact_1170_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_1171_append_Oright__neutral,axiom,
    ! [A: list_b] :
      ( ( append_b @ A @ nil_b )
      = A ) ).

% append.right_neutral
thf(fact_1172_append_Oright__neutral,axiom,
    ! [A: list_P8527749157015355191n_assn] :
      ( ( append282499809098378956n_assn @ A @ nil_Pr5671120429643327159n_assn )
      = A ) ).

% append.right_neutral
thf(fact_1173_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_1174_append_Oright__neutral,axiom,
    ! [A: list_int] :
      ( ( append_int @ A @ nil_int )
      = A ) ).

% append.right_neutral
thf(fact_1175_empty__append__eq__id,axiom,
    ( ( append_a @ nil_a )
    = ( ^ [X3: list_a] : X3 ) ) ).

% empty_append_eq_id
thf(fact_1176_empty__append__eq__id,axiom,
    ( ( append_b @ nil_b )
    = ( ^ [X3: list_b] : X3 ) ) ).

% empty_append_eq_id
thf(fact_1177_empty__append__eq__id,axiom,
    ( ( append282499809098378956n_assn @ nil_Pr5671120429643327159n_assn )
    = ( ^ [X3: list_P8527749157015355191n_assn] : X3 ) ) ).

% empty_append_eq_id
thf(fact_1178_empty__append__eq__id,axiom,
    ( ( append_nat @ nil_nat )
    = ( ^ [X3: list_nat] : X3 ) ) ).

% empty_append_eq_id
thf(fact_1179_empty__append__eq__id,axiom,
    ( ( append_int @ nil_int )
    = ( ^ [X3: list_int] : X3 ) ) ).

% empty_append_eq_id
thf(fact_1180_map__append,axiom,
    ! [F: nat > int,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_int @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_int @ ( map_nat_int @ F @ Xs ) @ ( map_nat_int @ F @ Ys ) ) ) ).

% map_append
thf(fact_1181_map__append,axiom,
    ! [F: int > nat,Xs: list_int,Ys: list_int] :
      ( ( map_int_nat @ F @ ( append_int @ Xs @ Ys ) )
      = ( append_nat @ ( map_int_nat @ F @ Xs ) @ ( map_int_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_1182_map__append,axiom,
    ! [F: int > int,Xs: list_int,Ys: list_int] :
      ( ( map_int_int @ F @ ( append_int @ Xs @ Ys ) )
      = ( append_int @ ( map_int_int @ F @ Xs ) @ ( map_int_int @ F @ Ys ) ) ) ).

% map_append
thf(fact_1183_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_1184_map__append,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn @ F @ ( append282499809098378956n_assn @ Xs @ Ys ) )
      = ( append_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) @ ( map_Pr8991440229025900053n_assn @ F @ Ys ) ) ) ).

% map_append
thf(fact_1185_rev__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( rev_a @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% rev_is_Nil_conv
thf(fact_1186_rev__is__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( ( rev_b @ Xs )
        = nil_b )
      = ( Xs = nil_b ) ) ).

% rev_is_Nil_conv
thf(fact_1187_rev__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( rev_Pr4855572775806611735n_assn @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rev_is_Nil_conv
thf(fact_1188_rev__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rev_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rev_is_Nil_conv
thf(fact_1189_rev__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rev_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rev_is_Nil_conv
thf(fact_1190_Nil__is__rev__conv,axiom,
    ! [Xs: list_a] :
      ( ( nil_a
        = ( rev_a @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_rev_conv
thf(fact_1191_Nil__is__rev__conv,axiom,
    ! [Xs: list_b] :
      ( ( nil_b
        = ( rev_b @ Xs ) )
      = ( Xs = nil_b ) ) ).

% Nil_is_rev_conv
thf(fact_1192_Nil__is__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( nil_Pr5671120429643327159n_assn
        = ( rev_Pr4855572775806611735n_assn @ Xs ) )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% Nil_is_rev_conv
thf(fact_1193_Nil__is__rev__conv,axiom,
    ! [Xs: list_nat] :
      ( ( nil_nat
        = ( rev_nat @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_rev_conv
thf(fact_1194_Nil__is__rev__conv,axiom,
    ! [Xs: list_int] :
      ( ( nil_int
        = ( rev_int @ Xs ) )
      = ( Xs = nil_int ) ) ).

% Nil_is_rev_conv
thf(fact_1195_rev__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( rev_nat @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( rev_nat @ Ys ) @ ( rev_nat @ Xs ) ) ) ).

% rev_append
thf(fact_1196_rev__append,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( rev_int @ ( append_int @ Xs @ Ys ) )
      = ( append_int @ ( rev_int @ Ys ) @ ( rev_int @ Xs ) ) ) ).

% rev_append
thf(fact_1197_lexordp__simps_I1_J,axiom,
    ! [Ys: list_nat] :
      ( ( ord_lexordp_nat @ nil_nat @ Ys )
      = ( Ys != nil_nat ) ) ).

% lexordp_simps(1)
thf(fact_1198_lexordp__simps_I1_J,axiom,
    ! [Ys: list_int] :
      ( ( ord_lexordp_int @ nil_int @ Ys )
      = ( Ys != nil_int ) ) ).

% lexordp_simps(1)
thf(fact_1199_lexordp__simps_I2_J,axiom,
    ! [Xs: list_nat] :
      ~ ( ord_lexordp_nat @ Xs @ nil_nat ) ).

% lexordp_simps(2)
thf(fact_1200_lexordp__simps_I2_J,axiom,
    ! [Xs: list_int] :
      ~ ( ord_lexordp_int @ Xs @ nil_int ) ).

% lexordp_simps(2)
thf(fact_1201_append1__eq__conv,axiom,
    ! [Xs: list_a,X: a,Ys: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
        = ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1202_append1__eq__conv,axiom,
    ! [Xs: list_b,X: b,Ys: list_b,Y: b] :
      ( ( ( append_b @ Xs @ ( cons_b @ X @ nil_b ) )
        = ( append_b @ Ys @ ( cons_b @ Y @ nil_b ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1203_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1204_append1__eq__conv,axiom,
    ! [Xs: list_int,X: int,Ys: list_int,Y: int] :
      ( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
        = ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1205_append1__eq__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn] :
      ( ( ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
        = ( append282499809098378956n_assn @ Ys @ ( cons_P2971678138204555879n_assn @ Y @ nil_Pr5671120429643327159n_assn ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1206_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_a,E2: a,L23: list_a,E3: a] :
      ( ( ( append_a @ L12 @ ( cons_a @ E2 @ L23 ) )
        = ( cons_a @ E3 @ nil_a ) )
      = ( ( L12 = nil_a )
        & ( E2 = E3 )
        & ( L23 = nil_a ) ) ) ).

% list_e_eq_lel(2)
thf(fact_1207_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_b,E2: b,L23: list_b,E3: b] :
      ( ( ( append_b @ L12 @ ( cons_b @ E2 @ L23 ) )
        = ( cons_b @ E3 @ nil_b ) )
      = ( ( L12 = nil_b )
        & ( E2 = E3 )
        & ( L23 = nil_b ) ) ) ).

% list_e_eq_lel(2)
thf(fact_1208_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_nat,E2: nat,L23: list_nat,E3: nat] :
      ( ( ( append_nat @ L12 @ ( cons_nat @ E2 @ L23 ) )
        = ( cons_nat @ E3 @ nil_nat ) )
      = ( ( L12 = nil_nat )
        & ( E2 = E3 )
        & ( L23 = nil_nat ) ) ) ).

% list_e_eq_lel(2)
thf(fact_1209_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_int,E2: int,L23: list_int,E3: int] :
      ( ( ( append_int @ L12 @ ( cons_int @ E2 @ L23 ) )
        = ( cons_int @ E3 @ nil_int ) )
      = ( ( L12 = nil_int )
        & ( E2 = E3 )
        & ( L23 = nil_int ) ) ) ).

% list_e_eq_lel(2)
thf(fact_1210_list__e__eq__lel_I2_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,E2: produc6575502325842934193n_assn,L23: list_P8527749157015355191n_assn,E3: produc6575502325842934193n_assn] :
      ( ( ( append282499809098378956n_assn @ L12 @ ( cons_P2971678138204555879n_assn @ E2 @ L23 ) )
        = ( cons_P2971678138204555879n_assn @ E3 @ nil_Pr5671120429643327159n_assn ) )
      = ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E2 = E3 )
        & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_e_eq_lel(2)
thf(fact_1211_list__e__eq__lel_I1_J,axiom,
    ! [E3: a,L12: list_a,E2: a,L23: list_a] :
      ( ( ( cons_a @ E3 @ nil_a )
        = ( append_a @ L12 @ ( cons_a @ E2 @ L23 ) ) )
      = ( ( L12 = nil_a )
        & ( E2 = E3 )
        & ( L23 = nil_a ) ) ) ).

% list_e_eq_lel(1)
thf(fact_1212_list__e__eq__lel_I1_J,axiom,
    ! [E3: b,L12: list_b,E2: b,L23: list_b] :
      ( ( ( cons_b @ E3 @ nil_b )
        = ( append_b @ L12 @ ( cons_b @ E2 @ L23 ) ) )
      = ( ( L12 = nil_b )
        & ( E2 = E3 )
        & ( L23 = nil_b ) ) ) ).

% list_e_eq_lel(1)
thf(fact_1213_list__e__eq__lel_I1_J,axiom,
    ! [E3: nat,L12: list_nat,E2: nat,L23: list_nat] :
      ( ( ( cons_nat @ E3 @ nil_nat )
        = ( append_nat @ L12 @ ( cons_nat @ E2 @ L23 ) ) )
      = ( ( L12 = nil_nat )
        & ( E2 = E3 )
        & ( L23 = nil_nat ) ) ) ).

% list_e_eq_lel(1)
thf(fact_1214_list__e__eq__lel_I1_J,axiom,
    ! [E3: int,L12: list_int,E2: int,L23: list_int] :
      ( ( ( cons_int @ E3 @ nil_int )
        = ( append_int @ L12 @ ( cons_int @ E2 @ L23 ) ) )
      = ( ( L12 = nil_int )
        & ( E2 = E3 )
        & ( L23 = nil_int ) ) ) ).

% list_e_eq_lel(1)
thf(fact_1215_list__e__eq__lel_I1_J,axiom,
    ! [E3: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn,E2: produc6575502325842934193n_assn,L23: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ E3 @ nil_Pr5671120429643327159n_assn )
        = ( append282499809098378956n_assn @ L12 @ ( cons_P2971678138204555879n_assn @ E2 @ L23 ) ) )
      = ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E2 = E3 )
        & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_e_eq_lel(1)
thf(fact_1216_list__se__match_I4_J,axiom,
    ! [L23: list_a,A: a,L12: list_a] :
      ( ( L23 != nil_a )
     => ( ( ( cons_a @ A @ nil_a )
          = ( append_a @ L12 @ L23 ) )
        = ( ( L12 = nil_a )
          & ( L23
            = ( cons_a @ A @ nil_a ) ) ) ) ) ).

% list_se_match(4)
thf(fact_1217_list__se__match_I4_J,axiom,
    ! [L23: list_b,A: b,L12: list_b] :
      ( ( L23 != nil_b )
     => ( ( ( cons_b @ A @ nil_b )
          = ( append_b @ L12 @ L23 ) )
        = ( ( L12 = nil_b )
          & ( L23
            = ( cons_b @ A @ nil_b ) ) ) ) ) ).

% list_se_match(4)
thf(fact_1218_list__se__match_I4_J,axiom,
    ! [L23: list_nat,A: nat,L12: list_nat] :
      ( ( L23 != nil_nat )
     => ( ( ( cons_nat @ A @ nil_nat )
          = ( append_nat @ L12 @ L23 ) )
        = ( ( L12 = nil_nat )
          & ( L23
            = ( cons_nat @ A @ nil_nat ) ) ) ) ) ).

% list_se_match(4)
thf(fact_1219_list__se__match_I4_J,axiom,
    ! [L23: list_int,A: int,L12: list_int] :
      ( ( L23 != nil_int )
     => ( ( ( cons_int @ A @ nil_int )
          = ( append_int @ L12 @ L23 ) )
        = ( ( L12 = nil_int )
          & ( L23
            = ( cons_int @ A @ nil_int ) ) ) ) ) ).

% list_se_match(4)
thf(fact_1220_list__se__match_I4_J,axiom,
    ! [L23: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn] :
      ( ( L23 != nil_Pr5671120429643327159n_assn )
     => ( ( ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn )
          = ( append282499809098378956n_assn @ L12 @ L23 ) )
        = ( ( L12 = nil_Pr5671120429643327159n_assn )
          & ( L23
            = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) ) ) ) ) ).

% list_se_match(4)
thf(fact_1221_list__se__match_I3_J,axiom,
    ! [L12: list_a,A: a,L23: list_a] :
      ( ( L12 != nil_a )
     => ( ( ( cons_a @ A @ nil_a )
          = ( append_a @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_a @ A @ nil_a ) )
          & ( L23 = nil_a ) ) ) ) ).

% list_se_match(3)
thf(fact_1222_list__se__match_I3_J,axiom,
    ! [L12: list_b,A: b,L23: list_b] :
      ( ( L12 != nil_b )
     => ( ( ( cons_b @ A @ nil_b )
          = ( append_b @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_b @ A @ nil_b ) )
          & ( L23 = nil_b ) ) ) ) ).

% list_se_match(3)
thf(fact_1223_list__se__match_I3_J,axiom,
    ! [L12: list_nat,A: nat,L23: list_nat] :
      ( ( L12 != nil_nat )
     => ( ( ( cons_nat @ A @ nil_nat )
          = ( append_nat @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_nat @ A @ nil_nat ) )
          & ( L23 = nil_nat ) ) ) ) ).

% list_se_match(3)
thf(fact_1224_list__se__match_I3_J,axiom,
    ! [L12: list_int,A: int,L23: list_int] :
      ( ( L12 != nil_int )
     => ( ( ( cons_int @ A @ nil_int )
          = ( append_int @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_int @ A @ nil_int ) )
          & ( L23 = nil_int ) ) ) ) ).

% list_se_match(3)
thf(fact_1225_list__se__match_I3_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn,L23: list_P8527749157015355191n_assn] :
      ( ( L12 != nil_Pr5671120429643327159n_assn )
     => ( ( ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn )
          = ( append282499809098378956n_assn @ L12 @ L23 ) )
        = ( ( L12
            = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) )
          & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ) ).

% list_se_match(3)
thf(fact_1226_list__se__match_I2_J,axiom,
    ! [L23: list_a,L12: list_a,A: a] :
      ( ( L23 != nil_a )
     => ( ( ( append_a @ L12 @ L23 )
          = ( cons_a @ A @ nil_a ) )
        = ( ( L12 = nil_a )
          & ( L23
            = ( cons_a @ A @ nil_a ) ) ) ) ) ).

% list_se_match(2)
thf(fact_1227_list__se__match_I2_J,axiom,
    ! [L23: list_b,L12: list_b,A: b] :
      ( ( L23 != nil_b )
     => ( ( ( append_b @ L12 @ L23 )
          = ( cons_b @ A @ nil_b ) )
        = ( ( L12 = nil_b )
          & ( L23
            = ( cons_b @ A @ nil_b ) ) ) ) ) ).

% list_se_match(2)
thf(fact_1228_list__se__match_I2_J,axiom,
    ! [L23: list_nat,L12: list_nat,A: nat] :
      ( ( L23 != nil_nat )
     => ( ( ( append_nat @ L12 @ L23 )
          = ( cons_nat @ A @ nil_nat ) )
        = ( ( L12 = nil_nat )
          & ( L23
            = ( cons_nat @ A @ nil_nat ) ) ) ) ) ).

% list_se_match(2)
thf(fact_1229_list__se__match_I2_J,axiom,
    ! [L23: list_int,L12: list_int,A: int] :
      ( ( L23 != nil_int )
     => ( ( ( append_int @ L12 @ L23 )
          = ( cons_int @ A @ nil_int ) )
        = ( ( L12 = nil_int )
          & ( L23
            = ( cons_int @ A @ nil_int ) ) ) ) ) ).

% list_se_match(2)
thf(fact_1230_list__se__match_I2_J,axiom,
    ! [L23: list_P8527749157015355191n_assn,L12: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( L23 != nil_Pr5671120429643327159n_assn )
     => ( ( ( append282499809098378956n_assn @ L12 @ L23 )
          = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) )
        = ( ( L12 = nil_Pr5671120429643327159n_assn )
          & ( L23
            = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) ) ) ) ) ).

% list_se_match(2)
thf(fact_1231_list__se__match_I1_J,axiom,
    ! [L12: list_a,L23: list_a,A: a] :
      ( ( L12 != nil_a )
     => ( ( ( append_a @ L12 @ L23 )
          = ( cons_a @ A @ nil_a ) )
        = ( ( L12
            = ( cons_a @ A @ nil_a ) )
          & ( L23 = nil_a ) ) ) ) ).

% list_se_match(1)
thf(fact_1232_list__se__match_I1_J,axiom,
    ! [L12: list_b,L23: list_b,A: b] :
      ( ( L12 != nil_b )
     => ( ( ( append_b @ L12 @ L23 )
          = ( cons_b @ A @ nil_b ) )
        = ( ( L12
            = ( cons_b @ A @ nil_b ) )
          & ( L23 = nil_b ) ) ) ) ).

% list_se_match(1)
thf(fact_1233_list__se__match_I1_J,axiom,
    ! [L12: list_nat,L23: list_nat,A: nat] :
      ( ( L12 != nil_nat )
     => ( ( ( append_nat @ L12 @ L23 )
          = ( cons_nat @ A @ nil_nat ) )
        = ( ( L12
            = ( cons_nat @ A @ nil_nat ) )
          & ( L23 = nil_nat ) ) ) ) ).

% list_se_match(1)
thf(fact_1234_list__se__match_I1_J,axiom,
    ! [L12: list_int,L23: list_int,A: int] :
      ( ( L12 != nil_int )
     => ( ( ( append_int @ L12 @ L23 )
          = ( cons_int @ A @ nil_int ) )
        = ( ( L12
            = ( cons_int @ A @ nil_int ) )
          & ( L23 = nil_int ) ) ) ) ).

% list_se_match(1)
thf(fact_1235_list__se__match_I1_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,L23: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( L12 != nil_Pr5671120429643327159n_assn )
     => ( ( ( append282499809098378956n_assn @ L12 @ L23 )
          = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) )
        = ( ( L12
            = ( cons_P2971678138204555879n_assn @ A @ nil_Pr5671120429643327159n_assn ) )
          & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ) ).

% list_se_match(1)
thf(fact_1236_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_a,E1: a,E22: a,L23: list_a,E12: a,E23: a] :
      ( ( ( append_a @ L12 @ ( cons_a @ E1 @ ( cons_a @ E22 @ L23 ) ) )
        = ( cons_a @ E12 @ ( cons_a @ E23 @ nil_a ) ) )
      = ( ( L12 = nil_a )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_a ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_1237_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_b,E1: b,E22: b,L23: list_b,E12: b,E23: b] :
      ( ( ( append_b @ L12 @ ( cons_b @ E1 @ ( cons_b @ E22 @ L23 ) ) )
        = ( cons_b @ E12 @ ( cons_b @ E23 @ nil_b ) ) )
      = ( ( L12 = nil_b )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_b ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_1238_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_nat,E1: nat,E22: nat,L23: list_nat,E12: nat,E23: nat] :
      ( ( ( append_nat @ L12 @ ( cons_nat @ E1 @ ( cons_nat @ E22 @ L23 ) ) )
        = ( cons_nat @ E12 @ ( cons_nat @ E23 @ nil_nat ) ) )
      = ( ( L12 = nil_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_nat ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_1239_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_int,E1: int,E22: int,L23: list_int,E12: int,E23: int] :
      ( ( ( append_int @ L12 @ ( cons_int @ E1 @ ( cons_int @ E22 @ L23 ) ) )
        = ( cons_int @ E12 @ ( cons_int @ E23 @ nil_int ) ) )
      = ( ( L12 = nil_int )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_int ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_1240_list__ee__eq__leel_I2_J,axiom,
    ! [L12: list_P8527749157015355191n_assn,E1: produc6575502325842934193n_assn,E22: produc6575502325842934193n_assn,L23: list_P8527749157015355191n_assn,E12: produc6575502325842934193n_assn,E23: produc6575502325842934193n_assn] :
      ( ( ( append282499809098378956n_assn @ L12 @ ( cons_P2971678138204555879n_assn @ E1 @ ( cons_P2971678138204555879n_assn @ E22 @ L23 ) ) )
        = ( cons_P2971678138204555879n_assn @ E12 @ ( cons_P2971678138204555879n_assn @ E23 @ nil_Pr5671120429643327159n_assn ) ) )
      = ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_ee_eq_leel(2)
thf(fact_1241_list__ee__eq__leel_I1_J,axiom,
    ! [E12: a,E23: a,L12: list_a,E1: a,E22: a,L23: list_a] :
      ( ( ( cons_a @ E12 @ ( cons_a @ E23 @ nil_a ) )
        = ( append_a @ L12 @ ( cons_a @ E1 @ ( cons_a @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_a )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_a ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_1242_list__ee__eq__leel_I1_J,axiom,
    ! [E12: b,E23: b,L12: list_b,E1: b,E22: b,L23: list_b] :
      ( ( ( cons_b @ E12 @ ( cons_b @ E23 @ nil_b ) )
        = ( append_b @ L12 @ ( cons_b @ E1 @ ( cons_b @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_b )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_b ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_1243_list__ee__eq__leel_I1_J,axiom,
    ! [E12: nat,E23: nat,L12: list_nat,E1: nat,E22: nat,L23: list_nat] :
      ( ( ( cons_nat @ E12 @ ( cons_nat @ E23 @ nil_nat ) )
        = ( append_nat @ L12 @ ( cons_nat @ E1 @ ( cons_nat @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_nat )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_nat ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_1244_list__ee__eq__leel_I1_J,axiom,
    ! [E12: int,E23: int,L12: list_int,E1: int,E22: int,L23: list_int] :
      ( ( ( cons_int @ E12 @ ( cons_int @ E23 @ nil_int ) )
        = ( append_int @ L12 @ ( cons_int @ E1 @ ( cons_int @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_int )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_int ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_1245_list__ee__eq__leel_I1_J,axiom,
    ! [E12: produc6575502325842934193n_assn,E23: produc6575502325842934193n_assn,L12: list_P8527749157015355191n_assn,E1: produc6575502325842934193n_assn,E22: produc6575502325842934193n_assn,L23: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ E12 @ ( cons_P2971678138204555879n_assn @ E23 @ nil_Pr5671120429643327159n_assn ) )
        = ( append282499809098378956n_assn @ L12 @ ( cons_P2971678138204555879n_assn @ E1 @ ( cons_P2971678138204555879n_assn @ E22 @ L23 ) ) ) )
      = ( ( L12 = nil_Pr5671120429643327159n_assn )
        & ( E12 = E1 )
        & ( E23 = E22 )
        & ( L23 = nil_Pr5671120429643327159n_assn ) ) ) ).

% list_ee_eq_leel(1)
thf(fact_1246_rev__singleton__conv,axiom,
    ! [Xs: list_a,X: a] :
      ( ( ( rev_a @ Xs )
        = ( cons_a @ X @ nil_a ) )
      = ( Xs
        = ( cons_a @ X @ nil_a ) ) ) ).

% rev_singleton_conv
thf(fact_1247_rev__singleton__conv,axiom,
    ! [Xs: list_b,X: b] :
      ( ( ( rev_b @ Xs )
        = ( cons_b @ X @ nil_b ) )
      = ( Xs
        = ( cons_b @ X @ nil_b ) ) ) ).

% rev_singleton_conv
thf(fact_1248_rev__singleton__conv,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( rev_nat @ Xs )
        = ( cons_nat @ X @ nil_nat ) )
      = ( Xs
        = ( cons_nat @ X @ nil_nat ) ) ) ).

% rev_singleton_conv
thf(fact_1249_rev__singleton__conv,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( rev_int @ Xs )
        = ( cons_int @ X @ nil_int ) )
      = ( Xs
        = ( cons_int @ X @ nil_int ) ) ) ).

% rev_singleton_conv
thf(fact_1250_rev__singleton__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ( rev_Pr4855572775806611735n_assn @ Xs )
        = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
      = ( Xs
        = ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ) ).

% rev_singleton_conv
thf(fact_1251_singleton__rev__conv,axiom,
    ! [X: a,Xs: list_a] :
      ( ( ( cons_a @ X @ nil_a )
        = ( rev_a @ Xs ) )
      = ( ( cons_a @ X @ nil_a )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_1252_singleton__rev__conv,axiom,
    ! [X: b,Xs: list_b] :
      ( ( ( cons_b @ X @ nil_b )
        = ( rev_b @ Xs ) )
      = ( ( cons_b @ X @ nil_b )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_1253_singleton__rev__conv,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( ( cons_nat @ X @ nil_nat )
        = ( rev_nat @ Xs ) )
      = ( ( cons_nat @ X @ nil_nat )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_1254_singleton__rev__conv,axiom,
    ! [X: int,Xs: list_int] :
      ( ( ( cons_int @ X @ nil_int )
        = ( rev_int @ Xs ) )
      = ( ( cons_int @ X @ nil_int )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_1255_singleton__rev__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn )
        = ( rev_Pr4855572775806611735n_assn @ Xs ) )
      = ( ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn )
        = Xs ) ) ).

% singleton_rev_conv
thf(fact_1256_lexordp__simps_I3_J,axiom,
    ! [X: assn,Xs: list_assn,Y: assn,Ys: list_assn] :
      ( ( ord_lexordp_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) )
      = ( ( ord_less_assn @ X @ Y )
        | ( ~ ( ord_less_assn @ Y @ X )
          & ( ord_lexordp_assn @ Xs @ Ys ) ) ) ) ).

% lexordp_simps(3)
thf(fact_1257_lexordp__simps_I3_J,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( ( ord_less_nat @ X @ Y )
        | ( ~ ( ord_less_nat @ Y @ X )
          & ( ord_lexordp_nat @ Xs @ Ys ) ) ) ) ).

% lexordp_simps(3)
thf(fact_1258_lexordp__simps_I3_J,axiom,
    ! [X: int,Xs: list_int,Y: int,Ys: list_int] :
      ( ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) )
      = ( ( ord_less_int @ X @ Y )
        | ( ~ ( ord_less_int @ Y @ X )
          & ( ord_lexordp_int @ Xs @ Ys ) ) ) ) ).

% lexordp_simps(3)
thf(fact_1259_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_int] :
      ( ~ ( finite_finite_int @ A3 )
     => ( ( linord2612477271533052124et_int @ A3 )
        = nil_int ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1260_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
    ! [A3: set_nat] :
      ( ~ ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ A3 )
        = nil_nat ) ) ).

% sorted_list_of_set.fold_insort_key.infinite
thf(fact_1261_list__collect__set__simps_I4_J,axiom,
    ! [F: nat > set_nat,L: list_nat,L3: list_nat] :
      ( ( list_c2452340269597857392at_nat @ F @ ( append_nat @ L @ L3 ) )
      = ( sup_sup_set_nat @ ( list_c2452340269597857392at_nat @ F @ L ) @ ( list_c2452340269597857392at_nat @ F @ L3 ) ) ) ).

% list_collect_set_simps(4)
thf(fact_1262_list__collect__set__simps_I4_J,axiom,
    ! [F: int > set_nat,L: list_int,L3: list_int] :
      ( ( list_c3451693981498911948nt_nat @ F @ ( append_int @ L @ L3 ) )
      = ( sup_sup_set_nat @ ( list_c3451693981498911948nt_nat @ F @ L ) @ ( list_c3451693981498911948nt_nat @ F @ L3 ) ) ) ).

% list_collect_set_simps(4)
thf(fact_1263_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: a > list_b] :
      ( ( bind_a_b @ ( cons_a @ X @ Xs ) @ F )
      = ( append_b @ ( F @ X ) @ ( bind_a_b @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1264_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: a > list_a] :
      ( ( bind_a_a @ ( cons_a @ X @ Xs ) @ F )
      = ( append_a @ ( F @ X ) @ ( bind_a_a @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1265_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: a > list_nat] :
      ( ( bind_a_nat @ ( cons_a @ X @ Xs ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_a_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1266_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: a > list_int] :
      ( ( bind_a_int @ ( cons_a @ X @ Xs ) @ F )
      = ( append_int @ ( F @ X ) @ ( bind_a_int @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1267_bind__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,F: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X @ Xs ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1268_bind__simps_I2_J,axiom,
    ! [X: nat,Xs: list_nat,F: nat > list_int] :
      ( ( bind_nat_int @ ( cons_nat @ X @ Xs ) @ F )
      = ( append_int @ ( F @ X ) @ ( bind_nat_int @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1269_bind__simps_I2_J,axiom,
    ! [X: int,Xs: list_int,F: int > list_nat] :
      ( ( bind_int_nat @ ( cons_int @ X @ Xs ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_int_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1270_bind__simps_I2_J,axiom,
    ! [X: int,Xs: list_int,F: int > list_int] :
      ( ( bind_int_int @ ( cons_int @ X @ Xs ) @ F )
      = ( append_int @ ( F @ X ) @ ( bind_int_int @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1271_bind__simps_I2_J,axiom,
    ! [X: a,Xs: list_a,F: a > list_P8527749157015355191n_assn] :
      ( ( bind_a3542047475819770682n_assn @ ( cons_a @ X @ Xs ) @ F )
      = ( append282499809098378956n_assn @ ( F @ X ) @ ( bind_a3542047475819770682n_assn @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1272_bind__simps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > list_nat] :
      ( ( bind_P8084169516273685562sn_nat @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ F )
      = ( append_nat @ ( F @ X ) @ ( bind_P8084169516273685562sn_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_1273_revg__fun,axiom,
    ( revg_nat
    = ( ^ [A2: list_nat] : ( append_nat @ ( rev_nat @ A2 ) ) ) ) ).

% revg_fun
thf(fact_1274_revg__fun,axiom,
    ( revg_int
    = ( ^ [A2: list_int] : ( append_int @ ( rev_int @ A2 ) ) ) ) ).

% revg_fun
thf(fact_1275_rev__eq__Cons__iff,axiom,
    ! [Xs: list_a,Y: a,Ys: list_a] :
      ( ( ( rev_a @ Xs )
        = ( cons_a @ Y @ Ys ) )
      = ( Xs
        = ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_1276_rev__eq__Cons__iff,axiom,
    ! [Xs: list_b,Y: b,Ys: list_b] :
      ( ( ( rev_b @ Xs )
        = ( cons_b @ Y @ Ys ) )
      = ( Xs
        = ( append_b @ ( rev_b @ Ys ) @ ( cons_b @ Y @ nil_b ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_1277_rev__eq__Cons__iff,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( rev_nat @ Xs )
        = ( cons_nat @ Y @ Ys ) )
      = ( Xs
        = ( append_nat @ ( rev_nat @ Ys ) @ ( cons_nat @ Y @ nil_nat ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_1278_rev__eq__Cons__iff,axiom,
    ! [Xs: list_int,Y: int,Ys: list_int] :
      ( ( ( rev_int @ Xs )
        = ( cons_int @ Y @ Ys ) )
      = ( Xs
        = ( append_int @ ( rev_int @ Ys ) @ ( cons_int @ Y @ nil_int ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_1279_rev__eq__Cons__iff,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Y: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( rev_Pr4855572775806611735n_assn @ Xs )
        = ( cons_P2971678138204555879n_assn @ Y @ Ys ) )
      = ( Xs
        = ( append282499809098378956n_assn @ ( rev_Pr4855572775806611735n_assn @ Ys ) @ ( cons_P2971678138204555879n_assn @ Y @ nil_Pr5671120429643327159n_assn ) ) ) ) ).

% rev_eq_Cons_iff
thf(fact_1280_psubsetD,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,C2: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2453136405763929_nat_o @ A3 @ B3 )
     => ( ( member6576561426505652726_nat_o @ C2 @ A3 )
       => ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% psubsetD
thf(fact_1281_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs3 @ Ts ) )
      = ( ? [Us: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs3 @ Us ) )
              & ( ( append_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us )
                = Zs3 )
              & ( Ys
                = ( append_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_1282_append__eq__append__conv2,axiom,
    ! [Xs: list_int,Ys: list_int,Zs3: list_int,Ts: list_int] :
      ( ( ( append_int @ Xs @ Ys )
        = ( append_int @ Zs3 @ Ts ) )
      = ( ? [Us: list_int] :
            ( ( ( Xs
                = ( append_int @ Zs3 @ Us ) )
              & ( ( append_int @ Us @ Ys )
                = Ts ) )
            | ( ( ( append_int @ Xs @ Us )
                = Zs3 )
              & ( Ys
                = ( append_int @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_1283_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs3: list_nat,Ys: list_nat,Us2: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us2 ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs3 @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_1284_append__eq__appendI,axiom,
    ! [Xs: list_int,Xs1: list_int,Zs3: list_int,Ys: list_int,Us2: list_int] :
      ( ( ( append_int @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys
          = ( append_int @ Xs1 @ Us2 ) )
       => ( ( append_int @ Xs @ Ys )
          = ( append_int @ Zs3 @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_1285_lexordp__append__leftI,axiom,
    ! [Us2: list_nat,Vs: list_nat,Xs: list_nat] :
      ( ( ord_lexordp_nat @ Us2 @ Vs )
     => ( ord_lexordp_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Xs @ Vs ) ) ) ).

% lexordp_append_leftI
thf(fact_1286_lexordp__append__leftI,axiom,
    ! [Us2: list_int,Vs: list_int,Xs: list_int] :
      ( ( ord_lexordp_int @ Us2 @ Vs )
     => ( ord_lexordp_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Xs @ Vs ) ) ) ).

% lexordp_append_leftI
thf(fact_1287_lexordp__append__leftD,axiom,
    ! [Xs: list_assn,Us2: list_assn,Vs: list_assn] :
      ( ( ord_lexordp_assn @ ( append_assn @ Xs @ Us2 ) @ ( append_assn @ Xs @ Vs ) )
     => ( ! [A4: assn] :
            ~ ( ord_less_assn @ A4 @ A4 )
       => ( ord_lexordp_assn @ Us2 @ Vs ) ) ) ).

% lexordp_append_leftD
thf(fact_1288_lexordp__append__leftD,axiom,
    ! [Xs: list_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ord_lexordp_nat @ ( append_nat @ Xs @ Us2 ) @ ( append_nat @ Xs @ Vs ) )
     => ( ! [A4: nat] :
            ~ ( ord_less_nat @ A4 @ A4 )
       => ( ord_lexordp_nat @ Us2 @ Vs ) ) ) ).

% lexordp_append_leftD
thf(fact_1289_lexordp__append__leftD,axiom,
    ! [Xs: list_int,Us2: list_int,Vs: list_int] :
      ( ( ord_lexordp_int @ ( append_int @ Xs @ Us2 ) @ ( append_int @ Xs @ Vs ) )
     => ( ! [A4: int] :
            ~ ( ord_less_int @ A4 @ A4 )
       => ( ord_lexordp_int @ Us2 @ Vs ) ) ) ).

% lexordp_append_leftD
thf(fact_1290_lexordp__append__rightI,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( ord_lexordp_nat @ Xs @ ( append_nat @ Xs @ Ys ) ) ) ).

% lexordp_append_rightI
thf(fact_1291_lexordp__append__rightI,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys != nil_int )
     => ( ord_lexordp_int @ Xs @ ( append_int @ Xs @ Ys ) ) ) ).

% lexordp_append_rightI
thf(fact_1292_rev_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( rev_a @ ( cons_a @ X @ Xs ) )
      = ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) ) ) ).

% rev.simps(2)
thf(fact_1293_rev_Osimps_I2_J,axiom,
    ! [X: b,Xs: list_b] :
      ( ( rev_b @ ( cons_b @ X @ Xs ) )
      = ( append_b @ ( rev_b @ Xs ) @ ( cons_b @ X @ nil_b ) ) ) ).

% rev.simps(2)
thf(fact_1294_rev_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( rev_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_nat @ ( rev_nat @ Xs ) @ ( cons_nat @ X @ nil_nat ) ) ) ).

% rev.simps(2)
thf(fact_1295_rev_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( rev_int @ ( cons_int @ X @ Xs ) )
      = ( append_int @ ( rev_int @ Xs ) @ ( cons_int @ X @ nil_int ) ) ) ).

% rev.simps(2)
thf(fact_1296_rev_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( rev_Pr4855572775806611735n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( append282499809098378956n_assn @ ( rev_Pr4855572775806611735n_assn @ Xs ) @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ) ).

% rev.simps(2)
thf(fact_1297_lexordp__iff,axiom,
    ( ord_lexordp_nat
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ? [X3: nat,Vs2: list_nat] :
              ( Ys3
              = ( append_nat @ Xs3 @ ( cons_nat @ X3 @ Vs2 ) ) )
          | ? [Us: list_nat,A2: nat,B2: nat,Vs2: list_nat,Ws: list_nat] :
              ( ( ord_less_nat @ A2 @ B2 )
              & ( Xs3
                = ( append_nat @ Us @ ( cons_nat @ A2 @ Vs2 ) ) )
              & ( Ys3
                = ( append_nat @ Us @ ( cons_nat @ B2 @ Ws ) ) ) ) ) ) ) ).

% lexordp_iff
thf(fact_1298_lexordp__iff,axiom,
    ( ord_lexordp_int
    = ( ^ [Xs3: list_int,Ys3: list_int] :
          ( ? [X3: int,Vs2: list_int] :
              ( Ys3
              = ( append_int @ Xs3 @ ( cons_int @ X3 @ Vs2 ) ) )
          | ? [Us: list_int,A2: int,B2: int,Vs2: list_int,Ws: list_int] :
              ( ( ord_less_int @ A2 @ B2 )
              & ( Xs3
                = ( append_int @ Us @ ( cons_int @ A2 @ Vs2 ) ) )
              & ( Ys3
                = ( append_int @ Us @ ( cons_int @ B2 @ Ws ) ) ) ) ) ) ) ).

% lexordp_iff
thf(fact_1299_lexordp__append__left__rightI,axiom,
    ! [X: assn,Y: assn,Us2: list_assn,Xs: list_assn,Ys: list_assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ord_lexordp_assn @ ( append_assn @ Us2 @ ( cons_assn @ X @ Xs ) ) @ ( append_assn @ Us2 @ ( cons_assn @ Y @ Ys ) ) ) ) ).

% lexordp_append_left_rightI
thf(fact_1300_lexordp__append__left__rightI,axiom,
    ! [X: nat,Y: nat,Us2: list_nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_lexordp_nat @ ( append_nat @ Us2 @ ( cons_nat @ X @ Xs ) ) @ ( append_nat @ Us2 @ ( cons_nat @ Y @ Ys ) ) ) ) ).

% lexordp_append_left_rightI
thf(fact_1301_lexordp__append__left__rightI,axiom,
    ! [X: int,Y: int,Us2: list_int,Xs: list_int,Ys: list_int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_lexordp_int @ ( append_int @ Us2 @ ( cons_int @ X @ Xs ) ) @ ( append_int @ Us2 @ ( cons_int @ Y @ Ys ) ) ) ) ).

% lexordp_append_left_rightI
thf(fact_1302_rev_Osimps_I1_J,axiom,
    ( ( rev_a @ nil_a )
    = nil_a ) ).

% rev.simps(1)
thf(fact_1303_rev_Osimps_I1_J,axiom,
    ( ( rev_b @ nil_b )
    = nil_b ) ).

% rev.simps(1)
thf(fact_1304_rev_Osimps_I1_J,axiom,
    ( ( rev_Pr4855572775806611735n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% rev.simps(1)
thf(fact_1305_rev_Osimps_I1_J,axiom,
    ( ( rev_nat @ nil_nat )
    = nil_nat ) ).

% rev.simps(1)
thf(fact_1306_rev_Osimps_I1_J,axiom,
    ( ( rev_int @ nil_int )
    = nil_int ) ).

% rev.simps(1)
thf(fact_1307_rev__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( rev_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( rev_nat @ Xs ) ) ) ).

% rev_map
thf(fact_1308_rev__map,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( rev_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) )
      = ( map_Pr8991440229025900053n_assn @ F @ ( rev_Pr4855572775806611735n_assn @ Xs ) ) ) ).

% rev_map
thf(fact_1309_append__Cons,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X @ Xs ) @ Ys )
      = ( cons_nat @ X @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1310_append__Cons,axiom,
    ! [X: int,Xs: list_int,Ys: list_int] :
      ( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
      = ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1311_append__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( append282499809098378956n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Ys )
      = ( cons_P2971678138204555879n_assn @ X @ ( append282499809098378956n_assn @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_1312_Cons__eq__appendI,axiom,
    ! [X: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_nat @ X @ Xs )
          = ( append_nat @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1313_Cons__eq__appendI,axiom,
    ! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs3: list_int] :
      ( ( ( cons_int @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_int @ Xs1 @ Zs3 ) )
       => ( ( cons_int @ X @ Xs )
          = ( append_int @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1314_Cons__eq__appendI,axiom,
    ! [X: produc6575502325842934193n_assn,Xs1: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append282499809098378956n_assn @ Xs1 @ Zs3 ) )
       => ( ( cons_P2971678138204555879n_assn @ X @ Xs )
          = ( append282499809098378956n_assn @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_1315_list__match__lel__lel,axiom,
    ! [C1: list_nat,Qs: nat,C22: list_nat,C12: list_nat,Qs2: nat,C23: list_nat] :
      ( ( ( append_nat @ C1 @ ( cons_nat @ Qs @ C22 ) )
        = ( append_nat @ C12 @ ( cons_nat @ Qs2 @ C23 ) ) )
     => ( ! [C21: list_nat] :
            ( ( C1
              = ( append_nat @ C12 @ ( cons_nat @ Qs2 @ C21 ) ) )
           => ( C23
             != ( append_nat @ C21 @ ( cons_nat @ Qs @ C22 ) ) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_nat] :
                ( ( C12
                  = ( append_nat @ C1 @ ( cons_nat @ Qs @ C212 ) ) )
               => ( C22
                 != ( append_nat @ C212 @ ( cons_nat @ Qs2 @ C23 ) ) ) ) ) ) ) ).

% list_match_lel_lel
thf(fact_1316_list__match__lel__lel,axiom,
    ! [C1: list_int,Qs: int,C22: list_int,C12: list_int,Qs2: int,C23: list_int] :
      ( ( ( append_int @ C1 @ ( cons_int @ Qs @ C22 ) )
        = ( append_int @ C12 @ ( cons_int @ Qs2 @ C23 ) ) )
     => ( ! [C21: list_int] :
            ( ( C1
              = ( append_int @ C12 @ ( cons_int @ Qs2 @ C21 ) ) )
           => ( C23
             != ( append_int @ C21 @ ( cons_int @ Qs @ C22 ) ) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_int] :
                ( ( C12
                  = ( append_int @ C1 @ ( cons_int @ Qs @ C212 ) ) )
               => ( C22
                 != ( append_int @ C212 @ ( cons_int @ Qs2 @ C23 ) ) ) ) ) ) ) ).

% list_match_lel_lel
thf(fact_1317_list__match__lel__lel,axiom,
    ! [C1: list_P8527749157015355191n_assn,Qs: produc6575502325842934193n_assn,C22: list_P8527749157015355191n_assn,C12: list_P8527749157015355191n_assn,Qs2: produc6575502325842934193n_assn,C23: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ C1 @ ( cons_P2971678138204555879n_assn @ Qs @ C22 ) )
        = ( append282499809098378956n_assn @ C12 @ ( cons_P2971678138204555879n_assn @ Qs2 @ C23 ) ) )
     => ( ! [C21: list_P8527749157015355191n_assn] :
            ( ( C1
              = ( append282499809098378956n_assn @ C12 @ ( cons_P2971678138204555879n_assn @ Qs2 @ C21 ) ) )
           => ( C23
             != ( append282499809098378956n_assn @ C21 @ ( cons_P2971678138204555879n_assn @ Qs @ C22 ) ) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list_P8527749157015355191n_assn] :
                ( ( C12
                  = ( append282499809098378956n_assn @ C1 @ ( cons_P2971678138204555879n_assn @ Qs @ C212 ) ) )
               => ( C22
                 != ( append282499809098378956n_assn @ C212 @ ( cons_P2971678138204555879n_assn @ Qs2 @ C23 ) ) ) ) ) ) ) ).

% list_match_lel_lel
thf(fact_1318_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_a @ nil_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1319_eq__Nil__appendI,axiom,
    ! [Xs: list_b,Ys: list_b] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_b @ nil_b @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1320_eq__Nil__appendI,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append282499809098378956n_assn @ nil_Pr5671120429643327159n_assn @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1321_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1322_eq__Nil__appendI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_int @ nil_int @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1323_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_1324_append_Oleft__neutral,axiom,
    ! [A: list_b] :
      ( ( append_b @ nil_b @ A )
      = A ) ).

% append.left_neutral
thf(fact_1325_append_Oleft__neutral,axiom,
    ! [A: list_P8527749157015355191n_assn] :
      ( ( append282499809098378956n_assn @ nil_Pr5671120429643327159n_assn @ A )
      = A ) ).

% append.left_neutral
thf(fact_1326_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_1327_append_Oleft__neutral,axiom,
    ! [A: list_int] :
      ( ( append_int @ nil_int @ A )
      = A ) ).

% append.left_neutral
thf(fact_1328_append__Nil,axiom,
    ! [Ys: list_a] :
      ( ( append_a @ nil_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1329_append__Nil,axiom,
    ! [Ys: list_b] :
      ( ( append_b @ nil_b @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1330_append__Nil,axiom,
    ! [Ys: list_P8527749157015355191n_assn] :
      ( ( append282499809098378956n_assn @ nil_Pr5671120429643327159n_assn @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1331_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1332_append__Nil,axiom,
    ! [Ys: list_int] :
      ( ( append_int @ nil_int @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1333_map__tailrec__rev,axiom,
    ( map_ta7164188454487880599at_nat
    = ( ^ [F3: nat > nat,As3: list_nat] : ( append_nat @ ( rev_nat @ ( map_nat_nat @ F3 @ As3 ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1334_map__tailrec__rev,axiom,
    ( map_ta5611102776663852196n_assn
    = ( ^ [F3: produc6575502325842934193n_assn > assn,As3: list_P8527749157015355191n_assn] : ( append_assn @ ( rev_assn @ ( map_Pr8991440229025900053n_assn @ F3 @ As3 ) ) ) ) ) ).

% map_tailrec_rev
thf(fact_1335_Misc_Omap__eq__append__conv,axiom,
    ! [F: int > nat,Ls: list_int,Fl: list_nat,Fl2: list_nat] :
      ( ( ( map_int_nat @ F @ Ls )
        = ( append_nat @ Fl @ Fl2 ) )
      = ( ? [L4: list_int,L5: list_int] :
            ( ( Ls
              = ( append_int @ L4 @ L5 ) )
            & ( ( map_int_nat @ F @ L4 )
              = Fl )
            & ( ( map_int_nat @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_1336_Misc_Omap__eq__append__conv,axiom,
    ! [F: nat > int,Ls: list_nat,Fl: list_int,Fl2: list_int] :
      ( ( ( map_nat_int @ F @ Ls )
        = ( append_int @ Fl @ Fl2 ) )
      = ( ? [L4: list_nat,L5: list_nat] :
            ( ( Ls
              = ( append_nat @ L4 @ L5 ) )
            & ( ( map_nat_int @ F @ L4 )
              = Fl )
            & ( ( map_nat_int @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_1337_Misc_Omap__eq__append__conv,axiom,
    ! [F: int > int,Ls: list_int,Fl: list_int,Fl2: list_int] :
      ( ( ( map_int_int @ F @ Ls )
        = ( append_int @ Fl @ Fl2 ) )
      = ( ? [L4: list_int,L5: list_int] :
            ( ( Ls
              = ( append_int @ L4 @ L5 ) )
            & ( ( map_int_int @ F @ L4 )
              = Fl )
            & ( ( map_int_int @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_1338_Misc_Omap__eq__append__conv,axiom,
    ! [F: nat > nat,Ls: list_nat,Fl: list_nat,Fl2: list_nat] :
      ( ( ( map_nat_nat @ F @ Ls )
        = ( append_nat @ Fl @ Fl2 ) )
      = ( ? [L4: list_nat,L5: list_nat] :
            ( ( Ls
              = ( append_nat @ L4 @ L5 ) )
            & ( ( map_nat_nat @ F @ L4 )
              = Fl )
            & ( ( map_nat_nat @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_1339_Misc_Omap__eq__append__conv,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Ls: list_P8527749157015355191n_assn,Fl: list_assn,Fl2: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Ls )
        = ( append_assn @ Fl @ Fl2 ) )
      = ( ? [L4: list_P8527749157015355191n_assn,L5: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( append282499809098378956n_assn @ L4 @ L5 ) )
            & ( ( map_Pr8991440229025900053n_assn @ F @ L4 )
              = Fl )
            & ( ( map_Pr8991440229025900053n_assn @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.map_eq_append_conv
thf(fact_1340_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: int > nat,Ls: list_int] :
      ( ( ( append_nat @ Fl @ Fl2 )
        = ( map_int_nat @ F @ Ls ) )
      = ( ? [L4: list_int,L5: list_int] :
            ( ( Ls
              = ( append_int @ L4 @ L5 ) )
            & ( ( map_int_nat @ F @ L4 )
              = Fl )
            & ( ( map_int_nat @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_1341_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_int,Fl2: list_int,F: nat > int,Ls: list_nat] :
      ( ( ( append_int @ Fl @ Fl2 )
        = ( map_nat_int @ F @ Ls ) )
      = ( ? [L4: list_nat,L5: list_nat] :
            ( ( Ls
              = ( append_nat @ L4 @ L5 ) )
            & ( ( map_nat_int @ F @ L4 )
              = Fl )
            & ( ( map_nat_int @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_1342_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_int,Fl2: list_int,F: int > int,Ls: list_int] :
      ( ( ( append_int @ Fl @ Fl2 )
        = ( map_int_int @ F @ Ls ) )
      = ( ? [L4: list_int,L5: list_int] :
            ( ( Ls
              = ( append_int @ L4 @ L5 ) )
            & ( ( map_int_int @ F @ L4 )
              = Fl )
            & ( ( map_int_int @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_1343_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: nat > nat,Ls: list_nat] :
      ( ( ( append_nat @ Fl @ Fl2 )
        = ( map_nat_nat @ F @ Ls ) )
      = ( ? [L4: list_nat,L5: list_nat] :
            ( ( Ls
              = ( append_nat @ L4 @ L5 ) )
            & ( ( map_nat_nat @ F @ L4 )
              = Fl )
            & ( ( map_nat_nat @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_1344_Misc_Oappend__eq__map__conv,axiom,
    ! [Fl: list_assn,Fl2: list_assn,F: produc6575502325842934193n_assn > assn,Ls: list_P8527749157015355191n_assn] :
      ( ( ( append_assn @ Fl @ Fl2 )
        = ( map_Pr8991440229025900053n_assn @ F @ Ls ) )
      = ( ? [L4: list_P8527749157015355191n_assn,L5: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( append282499809098378956n_assn @ L4 @ L5 ) )
            & ( ( map_Pr8991440229025900053n_assn @ F @ L4 )
              = Fl )
            & ( ( map_Pr8991440229025900053n_assn @ F @ L5 )
              = Fl2 ) ) ) ) ).

% Misc.append_eq_map_conv
thf(fact_1345_map__eq__appendE,axiom,
    ! [F: int > nat,Ls: list_int,Fl: list_nat,Fl2: list_nat] :
      ( ( ( map_int_nat @ F @ Ls )
        = ( append_nat @ Fl @ Fl2 ) )
     => ~ ! [L2: list_int,L6: list_int] :
            ( ( Ls
              = ( append_int @ L2 @ L6 ) )
           => ( ( ( map_int_nat @ F @ L2 )
                = Fl )
             => ( ( map_int_nat @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_1346_map__eq__appendE,axiom,
    ! [F: nat > int,Ls: list_nat,Fl: list_int,Fl2: list_int] :
      ( ( ( map_nat_int @ F @ Ls )
        = ( append_int @ Fl @ Fl2 ) )
     => ~ ! [L2: list_nat,L6: list_nat] :
            ( ( Ls
              = ( append_nat @ L2 @ L6 ) )
           => ( ( ( map_nat_int @ F @ L2 )
                = Fl )
             => ( ( map_nat_int @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_1347_map__eq__appendE,axiom,
    ! [F: int > int,Ls: list_int,Fl: list_int,Fl2: list_int] :
      ( ( ( map_int_int @ F @ Ls )
        = ( append_int @ Fl @ Fl2 ) )
     => ~ ! [L2: list_int,L6: list_int] :
            ( ( Ls
              = ( append_int @ L2 @ L6 ) )
           => ( ( ( map_int_int @ F @ L2 )
                = Fl )
             => ( ( map_int_int @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_1348_map__eq__appendE,axiom,
    ! [F: nat > nat,Ls: list_nat,Fl: list_nat,Fl2: list_nat] :
      ( ( ( map_nat_nat @ F @ Ls )
        = ( append_nat @ Fl @ Fl2 ) )
     => ~ ! [L2: list_nat,L6: list_nat] :
            ( ( Ls
              = ( append_nat @ L2 @ L6 ) )
           => ( ( ( map_nat_nat @ F @ L2 )
                = Fl )
             => ( ( map_nat_nat @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_1349_map__eq__appendE,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Ls: list_P8527749157015355191n_assn,Fl: list_assn,Fl2: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Ls )
        = ( append_assn @ Fl @ Fl2 ) )
     => ~ ! [L2: list_P8527749157015355191n_assn,L6: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( append282499809098378956n_assn @ L2 @ L6 ) )
           => ( ( ( map_Pr8991440229025900053n_assn @ F @ L2 )
                = Fl )
             => ( ( map_Pr8991440229025900053n_assn @ F @ L6 )
               != Fl2 ) ) ) ) ).

% map_eq_appendE
thf(fact_1350_append__eq__mapE,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: int > nat,Ls: list_int] :
      ( ( ( append_nat @ Fl @ Fl2 )
        = ( map_int_nat @ F @ Ls ) )
     => ~ ! [L2: list_int,L6: list_int] :
            ( ( Ls
              = ( append_int @ L2 @ L6 ) )
           => ( ( ( map_int_nat @ F @ L2 )
                = Fl )
             => ( ( map_int_nat @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_1351_append__eq__mapE,axiom,
    ! [Fl: list_int,Fl2: list_int,F: nat > int,Ls: list_nat] :
      ( ( ( append_int @ Fl @ Fl2 )
        = ( map_nat_int @ F @ Ls ) )
     => ~ ! [L2: list_nat,L6: list_nat] :
            ( ( Ls
              = ( append_nat @ L2 @ L6 ) )
           => ( ( ( map_nat_int @ F @ L2 )
                = Fl )
             => ( ( map_nat_int @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_1352_append__eq__mapE,axiom,
    ! [Fl: list_int,Fl2: list_int,F: int > int,Ls: list_int] :
      ( ( ( append_int @ Fl @ Fl2 )
        = ( map_int_int @ F @ Ls ) )
     => ~ ! [L2: list_int,L6: list_int] :
            ( ( Ls
              = ( append_int @ L2 @ L6 ) )
           => ( ( ( map_int_int @ F @ L2 )
                = Fl )
             => ( ( map_int_int @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_1353_append__eq__mapE,axiom,
    ! [Fl: list_nat,Fl2: list_nat,F: nat > nat,Ls: list_nat] :
      ( ( ( append_nat @ Fl @ Fl2 )
        = ( map_nat_nat @ F @ Ls ) )
     => ~ ! [L2: list_nat,L6: list_nat] :
            ( ( Ls
              = ( append_nat @ L2 @ L6 ) )
           => ( ( ( map_nat_nat @ F @ L2 )
                = Fl )
             => ( ( map_nat_nat @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_1354_append__eq__mapE,axiom,
    ! [Fl: list_assn,Fl2: list_assn,F: produc6575502325842934193n_assn > assn,Ls: list_P8527749157015355191n_assn] :
      ( ( ( append_assn @ Fl @ Fl2 )
        = ( map_Pr8991440229025900053n_assn @ F @ Ls ) )
     => ~ ! [L2: list_P8527749157015355191n_assn,L6: list_P8527749157015355191n_assn] :
            ( ( Ls
              = ( append282499809098378956n_assn @ L2 @ L6 ) )
           => ( ( ( map_Pr8991440229025900053n_assn @ F @ L2 )
                = Fl )
             => ( ( map_Pr8991440229025900053n_assn @ F @ L6 )
               != Fl2 ) ) ) ) ).

% append_eq_mapE
thf(fact_1355_List_Omap__eq__append__conv,axiom,
    ! [F: int > nat,Xs: list_int,Ys: list_nat,Zs3: list_nat] :
      ( ( ( map_int_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ? [Us: list_int,Vs2: list_int] :
            ( ( Xs
              = ( append_int @ Us @ Vs2 ) )
            & ( Ys
              = ( map_int_nat @ F @ Us ) )
            & ( Zs3
              = ( map_int_nat @ F @ Vs2 ) ) ) ) ) ).

% List.map_eq_append_conv
thf(fact_1356_List_Omap__eq__append__conv,axiom,
    ! [F: nat > int,Xs: list_nat,Ys: list_int,Zs3: list_int] :
      ( ( ( map_nat_int @ F @ Xs )
        = ( append_int @ Ys @ Zs3 ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_int @ F @ Us ) )
            & ( Zs3
              = ( map_nat_int @ F @ Vs2 ) ) ) ) ) ).

% List.map_eq_append_conv
thf(fact_1357_List_Omap__eq__append__conv,axiom,
    ! [F: int > int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( map_int_int @ F @ Xs )
        = ( append_int @ Ys @ Zs3 ) )
      = ( ? [Us: list_int,Vs2: list_int] :
            ( ( Xs
              = ( append_int @ Us @ Vs2 ) )
            & ( Ys
              = ( map_int_int @ F @ Us ) )
            & ( Zs3
              = ( map_int_int @ F @ Vs2 ) ) ) ) ) ).

% List.map_eq_append_conv
thf(fact_1358_List_Omap__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% List.map_eq_append_conv
thf(fact_1359_List_Omap__eq__append__conv,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,Ys: list_assn,Zs3: list_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( append_assn @ Ys @ Zs3 ) )
      = ( ? [Us: list_P8527749157015355191n_assn,Vs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( append282499809098378956n_assn @ Us @ Vs2 ) )
            & ( Ys
              = ( map_Pr8991440229025900053n_assn @ F @ Us ) )
            & ( Zs3
              = ( map_Pr8991440229025900053n_assn @ F @ Vs2 ) ) ) ) ) ).

% List.map_eq_append_conv
thf(fact_1360_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: int > nat,Xs: list_int] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( map_int_nat @ F @ Xs ) )
      = ( ? [Us: list_int,Vs2: list_int] :
            ( ( Xs
              = ( append_int @ Us @ Vs2 ) )
            & ( Ys
              = ( map_int_nat @ F @ Us ) )
            & ( Zs3
              = ( map_int_nat @ F @ Vs2 ) ) ) ) ) ).

% List.append_eq_map_conv
thf(fact_1361_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,F: nat > int,Xs: list_nat] :
      ( ( ( append_int @ Ys @ Zs3 )
        = ( map_nat_int @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_int @ F @ Us ) )
            & ( Zs3
              = ( map_nat_int @ F @ Vs2 ) ) ) ) ) ).

% List.append_eq_map_conv
thf(fact_1362_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,F: int > int,Xs: list_int] :
      ( ( ( append_int @ Ys @ Zs3 )
        = ( map_int_int @ F @ Xs ) )
      = ( ? [Us: list_int,Vs2: list_int] :
            ( ( Xs
              = ( append_int @ Us @ Vs2 ) )
            & ( Ys
              = ( map_int_int @ F @ Us ) )
            & ( Zs3
              = ( map_int_int @ F @ Vs2 ) ) ) ) ) ).

% List.append_eq_map_conv
thf(fact_1363_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% List.append_eq_map_conv
thf(fact_1364_List_Oappend__eq__map__conv,axiom,
    ! [Ys: list_assn,Zs3: list_assn,F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( append_assn @ Ys @ Zs3 )
        = ( map_Pr8991440229025900053n_assn @ F @ Xs ) )
      = ( ? [Us: list_P8527749157015355191n_assn,Vs2: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( append282499809098378956n_assn @ Us @ Vs2 ) )
            & ( Ys
              = ( map_Pr8991440229025900053n_assn @ F @ Us ) )
            & ( Zs3
              = ( map_Pr8991440229025900053n_assn @ F @ Vs2 ) ) ) ) ) ).

% List.append_eq_map_conv
thf(fact_1365_star__or__dist2,axiom,
    ! [C3: assn,A3: assn,B3: assn] :
      ( ( times_times_assn @ C3 @ ( sup_sup_assn @ A3 @ B3 ) )
      = ( sup_sup_assn @ ( times_times_assn @ C3 @ A3 ) @ ( times_times_assn @ C3 @ B3 ) ) ) ).

% star_or_dist2
thf(fact_1366_star__or__dist1,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( times_times_assn @ ( sup_sup_assn @ A3 @ B3 ) @ C3 )
      = ( sup_sup_assn @ ( times_times_assn @ A3 @ C3 ) @ ( times_times_assn @ B3 @ C3 ) ) ) ).

% star_or_dist1
thf(fact_1367_ord_Olexordp__eq__pref,axiom,
    ! [Less: nat > nat > $o,U: list_nat,V: list_nat] : ( lexordp_eq_nat @ Less @ U @ ( append_nat @ U @ V ) ) ).

% ord.lexordp_eq_pref
thf(fact_1368_ord_Olexordp__eq__pref,axiom,
    ! [Less: int > int > $o,U: list_int,V: list_int] : ( lexordp_eq_int @ Less @ U @ ( append_int @ U @ V ) ) ).

% ord.lexordp_eq_pref
thf(fact_1369_lexordp__irreflexive,axiom,
    ! [Xs: list_assn] :
      ( ! [X2: assn] :
          ~ ( ord_less_assn @ X2 @ X2 )
     => ~ ( ord_lexordp_assn @ Xs @ Xs ) ) ).

% lexordp_irreflexive
thf(fact_1370_lexordp__irreflexive,axiom,
    ! [Xs: list_nat] :
      ( ! [X2: nat] :
          ~ ( ord_less_nat @ X2 @ X2 )
     => ~ ( ord_lexordp_nat @ Xs @ Xs ) ) ).

% lexordp_irreflexive
thf(fact_1371_lexordp__irreflexive,axiom,
    ! [Xs: list_int] :
      ( ! [X2: int] :
          ~ ( ord_less_int @ X2 @ X2 )
     => ~ ( ord_lexordp_int @ Xs @ Xs ) ) ).

% lexordp_irreflexive
thf(fact_1372_norm__assertion__simps_I6_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ bot_bot_assn )
      = X ) ).

% norm_assertion_simps(6)
thf(fact_1373_norm__assertion__simps_I5_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ bot_bot_assn @ X )
      = X ) ).

% norm_assertion_simps(5)
thf(fact_1374_lexordp__eq__pref,axiom,
    ! [U: list_nat,V: list_nat] : ( ord_lexordp_eq_nat @ U @ ( append_nat @ U @ V ) ) ).

% lexordp_eq_pref
thf(fact_1375_lexordp__eq__pref,axiom,
    ! [U: list_int,V: list_int] : ( ord_lexordp_eq_int @ U @ ( append_int @ U @ V ) ) ).

% lexordp_eq_pref
thf(fact_1376_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( linord2614967742042102400et_nat @ A3 )
        = ( linord2614967742042102400et_nat @ B3 ) )
     => ( ( finite_finite_nat @ A3 )
       => ( ( finite_finite_nat @ B3 )
         => ( A3 = B3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_1377_merge_Osimps_I1_J,axiom,
    ! [L23: list_nat] :
      ( ( merge_nat @ nil_nat @ L23 )
      = L23 ) ).

% merge.simps(1)
thf(fact_1378_merge_Osimps_I1_J,axiom,
    ! [L23: list_int] :
      ( ( merge_int @ nil_int @ L23 )
      = L23 ) ).

% merge.simps(1)
thf(fact_1379_rev__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X2: a,Xs2: list_a] :
            ( ( P @ Xs2 )
           => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1380_rev__induct,axiom,
    ! [P: list_b > $o,Xs: list_b] :
      ( ( P @ nil_b )
     => ( ! [X2: b,Xs2: list_b] :
            ( ( P @ Xs2 )
           => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1381_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X2: nat,Xs2: list_nat] :
            ( ( P @ Xs2 )
           => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1382_rev__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ( P @ nil_int )
     => ( ! [X2: int,Xs2: list_int] :
            ( ( P @ Xs2 )
           => ( P @ ( append_int @ Xs2 @ ( cons_int @ X2 @ nil_int ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1383_rev__induct,axiom,
    ! [P: list_P8527749157015355191n_assn > $o,Xs: list_P8527749157015355191n_assn] :
      ( ( P @ nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
            ( ( P @ Xs2 )
           => ( P @ ( append282499809098378956n_assn @ Xs2 @ ( cons_P2971678138204555879n_assn @ X2 @ nil_Pr5671120429643327159n_assn ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1384_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys2: list_a,Y2: a] :
            ( Xs
           != ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_1385_rev__exhaust,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ~ ! [Ys2: list_b,Y2: b] :
            ( Xs
           != ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ).

% rev_exhaust
thf(fact_1386_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys2: list_nat,Y2: nat] :
            ( Xs
           != ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_1387_rev__exhaust,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ~ ! [Ys2: list_int,Y2: int] :
            ( Xs
           != ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).

% rev_exhaust
thf(fact_1388_rev__exhaust,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ~ ! [Ys2: list_P8527749157015355191n_assn,Y2: produc6575502325842934193n_assn] :
            ( Xs
           != ( append282499809098378956n_assn @ Ys2 @ ( cons_P2971678138204555879n_assn @ Y2 @ nil_Pr5671120429643327159n_assn ) ) ) ) ).

% rev_exhaust
thf(fact_1389_Cons__eq__append__conv,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( append_a @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_a )
          & ( ( cons_a @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_a] :
            ( ( ( cons_a @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_a @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1390_Cons__eq__append__conv,axiom,
    ! [X: b,Xs: list_b,Ys: list_b,Zs3: list_b] :
      ( ( ( cons_b @ X @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_b )
          & ( ( cons_b @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_b] :
            ( ( ( cons_b @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_b @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1391_Cons__eq__append__conv,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_nat] :
            ( ( ( cons_nat @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1392_Cons__eq__append__conv,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( append_int @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_int )
          & ( ( cons_int @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_int] :
            ( ( ( cons_int @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_int @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1393_Cons__eq__append__conv,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( append282499809098378956n_assn @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_Pr5671120429643327159n_assn )
          & ( ( cons_P2971678138204555879n_assn @ X @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_P8527749157015355191n_assn] :
            ( ( ( cons_P2971678138204555879n_assn @ X @ Ys5 )
              = Ys )
            & ( Xs
              = ( append282499809098378956n_assn @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_1394_append__eq__Cons__conv,axiom,
    ! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs3 )
        = ( cons_a @ X @ Xs ) )
      = ( ( ( Ys = nil_a )
          & ( Zs3
            = ( cons_a @ X @ Xs ) ) )
        | ? [Ys5: list_a] :
            ( ( Ys
              = ( cons_a @ X @ Ys5 ) )
            & ( ( append_a @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1395_append__eq__Cons__conv,axiom,
    ! [Ys: list_b,Zs3: list_b,X: b,Xs: list_b] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( cons_b @ X @ Xs ) )
      = ( ( ( Ys = nil_b )
          & ( Zs3
            = ( cons_b @ X @ Xs ) ) )
        | ? [Ys5: list_b] :
            ( ( Ys
              = ( cons_b @ X @ Ys5 ) )
            & ( ( append_b @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1396_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs3
            = ( cons_nat @ X @ Xs ) ) )
        | ? [Ys5: list_nat] :
            ( ( Ys
              = ( cons_nat @ X @ Ys5 ) )
            & ( ( append_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1397_append__eq__Cons__conv,axiom,
    ! [Ys: list_int,Zs3: list_int,X: int,Xs: list_int] :
      ( ( ( append_int @ Ys @ Zs3 )
        = ( cons_int @ X @ Xs ) )
      = ( ( ( Ys = nil_int )
          & ( Zs3
            = ( cons_int @ X @ Xs ) ) )
        | ? [Ys5: list_int] :
            ( ( Ys
              = ( cons_int @ X @ Ys5 ) )
            & ( ( append_int @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1398_append__eq__Cons__conv,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Ys @ Zs3 )
        = ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( ( ( Ys = nil_Pr5671120429643327159n_assn )
          & ( Zs3
            = ( cons_P2971678138204555879n_assn @ X @ Xs ) ) )
        | ? [Ys5: list_P8527749157015355191n_assn] :
            ( ( Ys
              = ( cons_P2971678138204555879n_assn @ X @ Ys5 ) )
            & ( ( append282499809098378956n_assn @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_1399_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X2: a] : ( P @ ( cons_a @ X2 @ nil_a ) )
       => ( ! [X2: a,Xs2: list_a] :
              ( ( Xs2 != nil_a )
             => ( ( P @ Xs2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1400_rev__nonempty__induct,axiom,
    ! [Xs: list_b,P: list_b > $o] :
      ( ( Xs != nil_b )
     => ( ! [X2: b] : ( P @ ( cons_b @ X2 @ nil_b ) )
       => ( ! [X2: b,Xs2: list_b] :
              ( ( Xs2 != nil_b )
             => ( ( P @ Xs2 )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1401_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs2: list_nat] :
              ( ( Xs2 != nil_nat )
             => ( ( P @ Xs2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1402_rev__nonempty__induct,axiom,
    ! [Xs: list_int,P: list_int > $o] :
      ( ( Xs != nil_int )
     => ( ! [X2: int] : ( P @ ( cons_int @ X2 @ nil_int ) )
       => ( ! [X2: int,Xs2: list_int] :
              ( ( Xs2 != nil_int )
             => ( ( P @ Xs2 )
               => ( P @ ( append_int @ Xs2 @ ( cons_int @ X2 @ nil_int ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1403_rev__nonempty__induct,axiom,
    ! [Xs: list_P8527749157015355191n_assn,P: list_P8527749157015355191n_assn > $o] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ! [X2: produc6575502325842934193n_assn] : ( P @ ( cons_P2971678138204555879n_assn @ X2 @ nil_Pr5671120429643327159n_assn ) )
       => ( ! [X2: produc6575502325842934193n_assn,Xs2: list_P8527749157015355191n_assn] :
              ( ( Xs2 != nil_Pr5671120429643327159n_assn )
             => ( ( P @ Xs2 )
               => ( P @ ( append282499809098378956n_assn @ Xs2 @ ( cons_P2971678138204555879n_assn @ X2 @ nil_Pr5671120429643327159n_assn ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_1404_neq__Nil__revE,axiom,
    ! [L: list_a] :
      ( ( L != nil_a )
     => ~ ! [Ll: list_a,E: a] :
            ( L
           != ( append_a @ Ll @ ( cons_a @ E @ nil_a ) ) ) ) ).

% neq_Nil_revE
thf(fact_1405_neq__Nil__revE,axiom,
    ! [L: list_b] :
      ( ( L != nil_b )
     => ~ ! [Ll: list_b,E: b] :
            ( L
           != ( append_b @ Ll @ ( cons_b @ E @ nil_b ) ) ) ) ).

% neq_Nil_revE
thf(fact_1406_neq__Nil__revE,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
     => ~ ! [Ll: list_nat,E: nat] :
            ( L
           != ( append_nat @ Ll @ ( cons_nat @ E @ nil_nat ) ) ) ) ).

% neq_Nil_revE
thf(fact_1407_neq__Nil__revE,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
     => ~ ! [Ll: list_int,E: int] :
            ( L
           != ( append_int @ Ll @ ( cons_int @ E @ nil_int ) ) ) ) ).

% neq_Nil_revE
thf(fact_1408_neq__Nil__revE,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
     => ~ ! [Ll: list_P8527749157015355191n_assn,E: produc6575502325842934193n_assn] :
            ( L
           != ( append282499809098378956n_assn @ Ll @ ( cons_P2971678138204555879n_assn @ E @ nil_Pr5671120429643327159n_assn ) ) ) ) ).

% neq_Nil_revE
thf(fact_1409_rev__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1410_rev__induct2_H,axiom,
    ! [P: list_a > list_b > $o,Xs: list_a,Ys: list_b] :
      ( ( P @ nil_a @ nil_b )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_a @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1411_rev__induct2_H,axiom,
    ! [P: list_b > list_a > $o,Xs: list_b,Ys: list_a] :
      ( ( P @ nil_b @ nil_a )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_b @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1412_rev__induct2_H,axiom,
    ! [P: list_b > list_b > $o,Xs: list_b,Ys: list_b] :
      ( ( P @ nil_b @ nil_b )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_b @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1413_rev__induct2_H,axiom,
    ! [P: list_a > list_nat > $o,Xs: list_a,Ys: list_nat] :
      ( ( P @ nil_a @ nil_nat )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_a @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1414_rev__induct2_H,axiom,
    ! [P: list_b > list_nat > $o,Xs: list_b,Ys: list_nat] :
      ( ( P @ nil_b @ nil_nat )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ nil_nat )
       => ( ! [Y2: nat,Ys2: list_nat] : ( P @ nil_b @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1415_rev__induct2_H,axiom,
    ! [P: list_a > list_int > $o,Xs: list_a,Ys: list_int] :
      ( ( P @ nil_a @ nil_int )
     => ( ! [X2: a,Xs2: list_a] : ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ nil_int )
       => ( ! [Y2: int,Ys2: list_int] : ( P @ nil_a @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) )
         => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1416_rev__induct2_H,axiom,
    ! [P: list_b > list_int > $o,Xs: list_b,Ys: list_int] :
      ( ( P @ nil_b @ nil_int )
     => ( ! [X2: b,Xs2: list_b] : ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ nil_int )
       => ( ! [Y2: int,Ys2: list_int] : ( P @ nil_b @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) )
         => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1417_rev__induct2_H,axiom,
    ! [P: list_nat > list_a > $o,Xs: list_nat,Ys: list_a] :
      ( ( P @ nil_nat @ nil_a )
     => ( ! [X2: nat,Xs2: list_nat] : ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_a )
       => ( ! [Y2: a,Ys2: list_a] : ( P @ nil_nat @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1418_rev__induct2_H,axiom,
    ! [P: list_nat > list_b > $o,Xs: list_nat,Ys: list_b] :
      ( ( P @ nil_nat @ nil_b )
     => ( ! [X2: nat,Xs2: list_nat] : ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ nil_b )
       => ( ! [Y2: b,Ys2: list_b] : ( P @ nil_nat @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                ( ( P @ Xs2 @ Ys2 )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% rev_induct2'
thf(fact_1419_neq__Nil__rev__conv,axiom,
    ! [L: list_a] :
      ( ( L != nil_a )
      = ( ? [Xs3: list_a,X3: a] :
            ( L
            = ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_1420_neq__Nil__rev__conv,axiom,
    ! [L: list_b] :
      ( ( L != nil_b )
      = ( ? [Xs3: list_b,X3: b] :
            ( L
            = ( append_b @ Xs3 @ ( cons_b @ X3 @ nil_b ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_1421_neq__Nil__rev__conv,axiom,
    ! [L: list_nat] :
      ( ( L != nil_nat )
      = ( ? [Xs3: list_nat,X3: nat] :
            ( L
            = ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_1422_neq__Nil__rev__conv,axiom,
    ! [L: list_int] :
      ( ( L != nil_int )
      = ( ? [Xs3: list_int,X3: int] :
            ( L
            = ( append_int @ Xs3 @ ( cons_int @ X3 @ nil_int ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_1423_neq__Nil__rev__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( L != nil_Pr5671120429643327159n_assn )
      = ( ? [Xs3: list_P8527749157015355191n_assn,X3: produc6575502325842934193n_assn] :
            ( L
            = ( append282499809098378956n_assn @ Xs3 @ ( cons_P2971678138204555879n_assn @ X3 @ nil_Pr5671120429643327159n_assn ) ) ) ) ) ).

% neq_Nil_rev_conv
thf(fact_1424_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_a )
       => ( ! [X2: a,Y2: a] : ( P @ ( cons_a @ X2 @ nil_a ) @ ( cons_a @ Y2 @ nil_a ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a] :
                ( ( Xs2 != nil_a )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( cons_a @ Y2 @ nil_a ) ) )
           => ( ! [X2: a,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => ( P @ ( cons_a @ X2 @ nil_a ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
             => ( ! [X2: a,Xs2: list_a,Y2: a,Ys2: list_a] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_a )
                       => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1425_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_b,P: list_a > list_b > $o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_b )
       => ( ! [X2: a,Y2: b] : ( P @ ( cons_a @ X2 @ nil_a ) @ ( cons_b @ Y2 @ nil_b ) )
         => ( ! [X2: a,Xs2: list_a,Y2: b] :
                ( ( Xs2 != nil_a )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( cons_b @ Y2 @ nil_b ) ) )
           => ( ! [X2: a,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => ( P @ ( cons_a @ X2 @ nil_a ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
             => ( ! [X2: a,Xs2: list_a,Y2: b,Ys2: list_b] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_b )
                       => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1426_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_a,P: list_b > list_a > $o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_a )
       => ( ! [X2: b,Y2: a] : ( P @ ( cons_b @ X2 @ nil_b ) @ ( cons_a @ Y2 @ nil_a ) )
         => ( ! [X2: b,Xs2: list_b,Y2: a] :
                ( ( Xs2 != nil_b )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( cons_a @ Y2 @ nil_a ) ) )
           => ( ! [X2: b,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => ( P @ ( cons_b @ X2 @ nil_b ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
             => ( ! [X2: b,Xs2: list_b,Y2: a,Ys2: list_a] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_a )
                       => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1427_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_b,P: list_b > list_b > $o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_b )
       => ( ! [X2: b,Y2: b] : ( P @ ( cons_b @ X2 @ nil_b ) @ ( cons_b @ Y2 @ nil_b ) )
         => ( ! [X2: b,Xs2: list_b,Y2: b] :
                ( ( Xs2 != nil_b )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( cons_b @ Y2 @ nil_b ) ) )
           => ( ! [X2: b,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => ( P @ ( cons_b @ X2 @ nil_b ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
             => ( ! [X2: b,Xs2: list_b,Y2: b,Ys2: list_b] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_b )
                       => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1428_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_nat,P: list_a > list_nat > $o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_nat )
       => ( ! [X2: a,Y2: nat] : ( P @ ( cons_a @ X2 @ nil_a ) @ ( cons_nat @ Y2 @ nil_nat ) )
         => ( ! [X2: a,Xs2: list_a,Y2: nat] :
                ( ( Xs2 != nil_a )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( cons_nat @ Y2 @ nil_nat ) ) )
           => ( ! [X2: a,Y2: nat,Ys2: list_nat] :
                  ( ( Ys2 != nil_nat )
                 => ( P @ ( cons_a @ X2 @ nil_a ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
             => ( ! [X2: a,Xs2: list_a,Y2: nat,Ys2: list_nat] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_nat )
                       => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1429_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_nat,P: list_b > list_nat > $o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_nat )
       => ( ! [X2: b,Y2: nat] : ( P @ ( cons_b @ X2 @ nil_b ) @ ( cons_nat @ Y2 @ nil_nat ) )
         => ( ! [X2: b,Xs2: list_b,Y2: nat] :
                ( ( Xs2 != nil_b )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( cons_nat @ Y2 @ nil_nat ) ) )
           => ( ! [X2: b,Y2: nat,Ys2: list_nat] :
                  ( ( Ys2 != nil_nat )
                 => ( P @ ( cons_b @ X2 @ nil_b ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) )
             => ( ! [X2: b,Xs2: list_b,Y2: nat,Ys2: list_nat] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_nat )
                       => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1430_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_a,Ys: list_int,P: list_a > list_int > $o] :
      ( ( Xs != nil_a )
     => ( ( Ys != nil_int )
       => ( ! [X2: a,Y2: int] : ( P @ ( cons_a @ X2 @ nil_a ) @ ( cons_int @ Y2 @ nil_int ) )
         => ( ! [X2: a,Xs2: list_a,Y2: int] :
                ( ( Xs2 != nil_a )
               => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( cons_int @ Y2 @ nil_int ) ) )
           => ( ! [X2: a,Y2: int,Ys2: list_int] :
                  ( ( Ys2 != nil_int )
                 => ( P @ ( cons_a @ X2 @ nil_a ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) )
             => ( ! [X2: a,Xs2: list_a,Y2: int,Ys2: list_int] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_a )
                     => ( ( Ys2 != nil_int )
                       => ( P @ ( append_a @ Xs2 @ ( cons_a @ X2 @ nil_a ) ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1431_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_b,Ys: list_int,P: list_b > list_int > $o] :
      ( ( Xs != nil_b )
     => ( ( Ys != nil_int )
       => ( ! [X2: b,Y2: int] : ( P @ ( cons_b @ X2 @ nil_b ) @ ( cons_int @ Y2 @ nil_int ) )
         => ( ! [X2: b,Xs2: list_b,Y2: int] :
                ( ( Xs2 != nil_b )
               => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( cons_int @ Y2 @ nil_int ) ) )
           => ( ! [X2: b,Y2: int,Ys2: list_int] :
                  ( ( Ys2 != nil_int )
                 => ( P @ ( cons_b @ X2 @ nil_b ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) )
             => ( ! [X2: b,Xs2: list_b,Y2: int,Ys2: list_int] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_b )
                     => ( ( Ys2 != nil_int )
                       => ( P @ ( append_b @ Xs2 @ ( cons_b @ X2 @ nil_b ) ) @ ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1432_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_nat,Ys: list_a,P: list_nat > list_a > $o] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_a )
       => ( ! [X2: nat,Y2: a] : ( P @ ( cons_nat @ X2 @ nil_nat ) @ ( cons_a @ Y2 @ nil_a ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: a] :
                ( ( Xs2 != nil_nat )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( cons_a @ Y2 @ nil_a ) ) )
           => ( ! [X2: nat,Y2: a,Ys2: list_a] :
                  ( ( Ys2 != nil_a )
                 => ( P @ ( cons_nat @ X2 @ nil_nat ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) )
             => ( ! [X2: nat,Xs2: list_nat,Y2: a,Ys2: list_a] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_nat )
                     => ( ( Ys2 != nil_a )
                       => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1433_rev__nonempty__induct2_H,axiom,
    ! [Xs: list_nat,Ys: list_b,P: list_nat > list_b > $o] :
      ( ( Xs != nil_nat )
     => ( ( Ys != nil_b )
       => ( ! [X2: nat,Y2: b] : ( P @ ( cons_nat @ X2 @ nil_nat ) @ ( cons_b @ Y2 @ nil_b ) )
         => ( ! [X2: nat,Xs2: list_nat,Y2: b] :
                ( ( Xs2 != nil_nat )
               => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( cons_b @ Y2 @ nil_b ) ) )
           => ( ! [X2: nat,Y2: b,Ys2: list_b] :
                  ( ( Ys2 != nil_b )
                 => ( P @ ( cons_nat @ X2 @ nil_nat ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) )
             => ( ! [X2: nat,Xs2: list_nat,Y2: b,Ys2: list_b] :
                    ( ( P @ Xs2 @ Ys2 )
                   => ( ( Xs2 != nil_nat )
                     => ( ( Ys2 != nil_b )
                       => ( P @ ( append_nat @ Xs2 @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_b @ Ys2 @ ( cons_b @ Y2 @ nil_b ) ) ) ) ) )
               => ( P @ Xs @ Ys ) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
thf(fact_1434_list__Cons__eq__append__cases,axiom,
    ! [X: a,Xs: list_a,Ys: list_a,Zs3: list_a] :
      ( ( ( cons_a @ X @ Xs )
        = ( append_a @ Ys @ Zs3 ) )
     => ( ( ( Ys = nil_a )
         => ( Zs3
           != ( cons_a @ X @ Xs ) ) )
       => ~ ! [Ys4: list_a] :
              ( ( Ys
                = ( cons_a @ X @ Ys4 ) )
             => ( ( append_a @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_1435_list__Cons__eq__append__cases,axiom,
    ! [X: b,Xs: list_b,Ys: list_b,Zs3: list_b] :
      ( ( ( cons_b @ X @ Xs )
        = ( append_b @ Ys @ Zs3 ) )
     => ( ( ( Ys = nil_b )
         => ( Zs3
           != ( cons_b @ X @ Xs ) ) )
       => ~ ! [Ys4: list_b] :
              ( ( Ys
                = ( cons_b @ X @ Ys4 ) )
             => ( ( append_b @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_1436_list__Cons__eq__append__cases,axiom,
    ! [X: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
     => ( ( ( Ys = nil_nat )
         => ( Zs3
           != ( cons_nat @ X @ Xs ) ) )
       => ~ ! [Ys4: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Ys4 ) )
             => ( ( append_nat @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_1437_list__Cons__eq__append__cases,axiom,
    ! [X: int,Xs: list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( cons_int @ X @ Xs )
        = ( append_int @ Ys @ Zs3 ) )
     => ( ( ( Ys = nil_int )
         => ( Zs3
           != ( cons_int @ X @ Xs ) ) )
       => ~ ! [Ys4: list_int] :
              ( ( Ys
                = ( cons_int @ X @ Ys4 ) )
             => ( ( append_int @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_1438_list__Cons__eq__append__cases,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( ( cons_P2971678138204555879n_assn @ X @ Xs )
        = ( append282499809098378956n_assn @ Ys @ Zs3 ) )
     => ( ( ( Ys = nil_Pr5671120429643327159n_assn )
         => ( Zs3
           != ( cons_P2971678138204555879n_assn @ X @ Xs ) ) )
       => ~ ! [Ys4: list_P8527749157015355191n_assn] :
              ( ( Ys
                = ( cons_P2971678138204555879n_assn @ X @ Ys4 ) )
             => ( ( append282499809098378956n_assn @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
thf(fact_1439_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_a,Zs3: list_a,X: a,Xs: list_a] :
      ( ( ( append_a @ Ys @ Zs3 )
        = ( cons_a @ X @ Xs ) )
     => ( ( ( Ys = nil_a )
         => ( Zs3
           != ( cons_a @ X @ Xs ) ) )
       => ~ ! [Ys4: list_a] :
              ( ( Ys
                = ( cons_a @ X @ Ys4 ) )
             => ( ( append_a @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_1440_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_b,Zs3: list_b,X: b,Xs: list_b] :
      ( ( ( append_b @ Ys @ Zs3 )
        = ( cons_b @ X @ Xs ) )
     => ( ( ( Ys = nil_b )
         => ( Zs3
           != ( cons_b @ X @ Xs ) ) )
       => ~ ! [Ys4: list_b] :
              ( ( Ys
                = ( cons_b @ X @ Ys4 ) )
             => ( ( append_b @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_1441_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_nat,Zs3: list_nat,X: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( cons_nat @ X @ Xs ) )
     => ( ( ( Ys = nil_nat )
         => ( Zs3
           != ( cons_nat @ X @ Xs ) ) )
       => ~ ! [Ys4: list_nat] :
              ( ( Ys
                = ( cons_nat @ X @ Ys4 ) )
             => ( ( append_nat @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_1442_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_int,Zs3: list_int,X: int,Xs: list_int] :
      ( ( ( append_int @ Ys @ Zs3 )
        = ( cons_int @ X @ Xs ) )
     => ( ( ( Ys = nil_int )
         => ( Zs3
           != ( cons_int @ X @ Xs ) ) )
       => ~ ! [Ys4: list_int] :
              ( ( Ys
                = ( cons_int @ X @ Ys4 ) )
             => ( ( append_int @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_1443_list__append__eq__Cons__cases,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Ys @ Zs3 )
        = ( cons_P2971678138204555879n_assn @ X @ Xs ) )
     => ( ( ( Ys = nil_Pr5671120429643327159n_assn )
         => ( Zs3
           != ( cons_P2971678138204555879n_assn @ X @ Xs ) ) )
       => ~ ! [Ys4: list_P8527749157015355191n_assn] :
              ( ( Ys
                = ( cons_P2971678138204555879n_assn @ X @ Ys4 ) )
             => ( ( append282499809098378956n_assn @ Ys4 @ Zs3 )
               != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
thf(fact_1444_map__consI_I2_J,axiom,
    ! [W: list_assn,L: list_assn,F: produc6575502325842934193n_assn > assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( ( append_assn @ W @ L )
        = ( append_assn @ ( map_Pr8991440229025900053n_assn @ F @ Ww ) @ L ) )
     => ( ( cons_assn @ ( F @ A ) @ ( append_assn @ W @ L ) )
        = ( append_assn @ ( map_Pr8991440229025900053n_assn @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1445_map__consI_I2_J,axiom,
    ! [W: list_nat,L: list_nat,F: nat > nat,Ww: list_nat,A: nat] :
      ( ( ( append_nat @ W @ L )
        = ( append_nat @ ( map_nat_nat @ F @ Ww ) @ L ) )
     => ( ( cons_nat @ ( F @ A ) @ ( append_nat @ W @ L ) )
        = ( append_nat @ ( map_nat_nat @ F @ ( cons_nat @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1446_map__consI_I2_J,axiom,
    ! [W: list_nat,L: list_nat,F: int > nat,Ww: list_int,A: int] :
      ( ( ( append_nat @ W @ L )
        = ( append_nat @ ( map_int_nat @ F @ Ww ) @ L ) )
     => ( ( cons_nat @ ( F @ A ) @ ( append_nat @ W @ L ) )
        = ( append_nat @ ( map_int_nat @ F @ ( cons_int @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1447_map__consI_I2_J,axiom,
    ! [W: list_nat,L: list_nat,F: produc6575502325842934193n_assn > nat,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( ( append_nat @ W @ L )
        = ( append_nat @ ( map_Pr7570552894071451325sn_nat @ F @ Ww ) @ L ) )
     => ( ( cons_nat @ ( F @ A ) @ ( append_nat @ W @ L ) )
        = ( append_nat @ ( map_Pr7570552894071451325sn_nat @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1448_map__consI_I2_J,axiom,
    ! [W: list_int,L: list_int,F: nat > int,Ww: list_nat,A: nat] :
      ( ( ( append_int @ W @ L )
        = ( append_int @ ( map_nat_int @ F @ Ww ) @ L ) )
     => ( ( cons_int @ ( F @ A ) @ ( append_int @ W @ L ) )
        = ( append_int @ ( map_nat_int @ F @ ( cons_nat @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1449_map__consI_I2_J,axiom,
    ! [W: list_int,L: list_int,F: int > int,Ww: list_int,A: int] :
      ( ( ( append_int @ W @ L )
        = ( append_int @ ( map_int_int @ F @ Ww ) @ L ) )
     => ( ( cons_int @ ( F @ A ) @ ( append_int @ W @ L ) )
        = ( append_int @ ( map_int_int @ F @ ( cons_int @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1450_map__consI_I2_J,axiom,
    ! [W: list_int,L: list_int,F: produc6575502325842934193n_assn > int,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( ( append_int @ W @ L )
        = ( append_int @ ( map_Pr7568062423562401049sn_int @ F @ Ww ) @ L ) )
     => ( ( cons_int @ ( F @ A ) @ ( append_int @ W @ L ) )
        = ( append_int @ ( map_Pr7568062423562401049sn_int @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1451_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,L: list_P8527749157015355191n_assn,F: nat > produc6575502325842934193n_assn,Ww: list_nat,A: nat] :
      ( ( ( append282499809098378956n_assn @ W @ L )
        = ( append282499809098378956n_assn @ ( map_na2667955367175718043n_assn @ F @ Ww ) @ L ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ ( append282499809098378956n_assn @ W @ L ) )
        = ( append282499809098378956n_assn @ ( map_na2667955367175718043n_assn @ F @ ( cons_nat @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1452_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,L: list_P8527749157015355191n_assn,F: int > produc6575502325842934193n_assn,Ww: list_int,A: int] :
      ( ( ( append282499809098378956n_assn @ W @ L )
        = ( append282499809098378956n_assn @ ( map_in4427992030928829247n_assn @ F @ Ww ) @ L ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ ( append282499809098378956n_assn @ W @ L ) )
        = ( append282499809098378956n_assn @ ( map_in4427992030928829247n_assn @ F @ ( cons_int @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1453_map__consI_I2_J,axiom,
    ! [W: list_P8527749157015355191n_assn,L: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > produc6575502325842934193n_assn,Ww: list_P8527749157015355191n_assn,A: produc6575502325842934193n_assn] :
      ( ( ( append282499809098378956n_assn @ W @ L )
        = ( append282499809098378956n_assn @ ( map_Pr7925354932063753860n_assn @ F @ Ww ) @ L ) )
     => ( ( cons_P2971678138204555879n_assn @ ( F @ A ) @ ( append282499809098378956n_assn @ W @ L ) )
        = ( append282499809098378956n_assn @ ( map_Pr7925354932063753860n_assn @ F @ ( cons_P2971678138204555879n_assn @ A @ Ww ) ) @ L ) ) ) ).

% map_consI(2)
thf(fact_1454_SLN__left,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ sln @ P )
      = P ) ).

% SLN_left
thf(fact_1455_SLN__right,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ P @ sln )
      = P ) ).

% SLN_right
thf(fact_1456_lexordp_OCons,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ord_lexordp_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) ) ) ).

% lexordp.Cons
thf(fact_1457_lexordp_OCons,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ).

% lexordp.Cons
thf(fact_1458_lexordp_OCons,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ).

% lexordp.Cons
thf(fact_1459_lexordp_OCons__eq,axiom,
    ! [X: assn,Y: assn,Xs: list_assn,Ys: list_assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ~ ( ord_less_assn @ Y @ X )
       => ( ( ord_lexordp_assn @ Xs @ Ys )
         => ( ord_lexordp_assn @ ( cons_assn @ X @ Xs ) @ ( cons_assn @ Y @ Ys ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_1460_lexordp_OCons__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ~ ( ord_less_nat @ Y @ X )
       => ( ( ord_lexordp_nat @ Xs @ Ys )
         => ( ord_lexordp_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_1461_lexordp_OCons__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,Ys: list_int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ~ ( ord_less_int @ Y @ X )
       => ( ( ord_lexordp_int @ Xs @ Ys )
         => ( ord_lexordp_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) ) ) ) ).

% lexordp.Cons_eq
thf(fact_1462_lexordp_ONil,axiom,
    ! [Y: nat,Ys: list_nat] : ( ord_lexordp_nat @ nil_nat @ ( cons_nat @ Y @ Ys ) ) ).

% lexordp.Nil
thf(fact_1463_lexordp_ONil,axiom,
    ! [Y: int,Ys: list_int] : ( ord_lexordp_int @ nil_int @ ( cons_int @ Y @ Ys ) ) ).

% lexordp.Nil
thf(fact_1464_SLN__def,axiom,
    sln = one_one_assn ).

% SLN_def
thf(fact_1465_merge_Osimps_I3_J,axiom,
    ! [X12: nat,X24: nat,L12: list_nat,L23: list_nat] :
      ( ( ( ord_less_nat @ X12 @ X24 )
       => ( ( merge_nat @ ( cons_nat @ X12 @ L12 ) @ ( cons_nat @ X24 @ L23 ) )
          = ( cons_nat @ X12 @ ( merge_nat @ L12 @ ( cons_nat @ X24 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_nat @ X12 @ X24 )
       => ( ( ( X12 = X24 )
           => ( ( merge_nat @ ( cons_nat @ X12 @ L12 ) @ ( cons_nat @ X24 @ L23 ) )
              = ( cons_nat @ X12 @ ( merge_nat @ L12 @ L23 ) ) ) )
          & ( ( X12 != X24 )
           => ( ( merge_nat @ ( cons_nat @ X12 @ L12 ) @ ( cons_nat @ X24 @ L23 ) )
              = ( cons_nat @ X24 @ ( merge_nat @ ( cons_nat @ X12 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_1466_merge_Osimps_I3_J,axiom,
    ! [X12: int,X24: int,L12: list_int,L23: list_int] :
      ( ( ( ord_less_int @ X12 @ X24 )
       => ( ( merge_int @ ( cons_int @ X12 @ L12 ) @ ( cons_int @ X24 @ L23 ) )
          = ( cons_int @ X12 @ ( merge_int @ L12 @ ( cons_int @ X24 @ L23 ) ) ) ) )
      & ( ~ ( ord_less_int @ X12 @ X24 )
       => ( ( ( X12 = X24 )
           => ( ( merge_int @ ( cons_int @ X12 @ L12 ) @ ( cons_int @ X24 @ L23 ) )
              = ( cons_int @ X12 @ ( merge_int @ L12 @ L23 ) ) ) )
          & ( ( X12 != X24 )
           => ( ( merge_int @ ( cons_int @ X12 @ L12 ) @ ( cons_int @ X24 @ L23 ) )
              = ( cons_int @ X24 @ ( merge_int @ ( cons_int @ X12 @ L12 ) @ L23 ) ) ) ) ) ) ) ).

% merge.simps(3)
thf(fact_1467_merge_Osimps_I2_J,axiom,
    ! [V: nat,Va: list_nat] :
      ( ( merge_nat @ ( cons_nat @ V @ Va ) @ nil_nat )
      = ( cons_nat @ V @ Va ) ) ).

% merge.simps(2)
thf(fact_1468_merge_Osimps_I2_J,axiom,
    ! [V: int,Va: list_int] :
      ( ( merge_int @ ( cons_int @ V @ Va ) @ nil_int )
      = ( cons_int @ V @ Va ) ) ).

% merge.simps(2)
thf(fact_1469_lexordp_Ocases,axiom,
    ! [A1: list_assn,A22: list_assn] :
      ( ( ord_lexordp_assn @ A1 @ A22 )
     => ( ( ( A1 = nil_assn )
         => ! [Y2: assn,Ys2: list_assn] :
              ( A22
             != ( cons_assn @ Y2 @ Ys2 ) ) )
       => ( ! [X2: assn] :
              ( ? [Xs2: list_assn] :
                  ( A1
                  = ( cons_assn @ X2 @ Xs2 ) )
             => ! [Y2: assn] :
                  ( ? [Ys2: list_assn] :
                      ( A22
                      = ( cons_assn @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_assn @ X2 @ Y2 ) ) )
         => ~ ! [X2: assn,Y2: assn,Xs2: list_assn] :
                ( ( A1
                  = ( cons_assn @ X2 @ Xs2 ) )
               => ! [Ys2: list_assn] :
                    ( ( A22
                      = ( cons_assn @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_assn @ X2 @ Y2 )
                     => ( ~ ( ord_less_assn @ Y2 @ X2 )
                       => ~ ( ord_lexordp_assn @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_1470_lexordp_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat] :
      ( ( ord_lexordp_nat @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ! [Y2: nat,Ys2: list_nat] :
              ( A22
             != ( cons_nat @ Y2 @ Ys2 ) ) )
       => ( ! [X2: nat] :
              ( ? [Xs2: list_nat] :
                  ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
             => ! [Y2: nat] :
                  ( ? [Ys2: list_nat] :
                      ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_nat @ X2 @ Y2 ) ) )
         => ~ ! [X2: nat,Y2: nat,Xs2: list_nat] :
                ( ( A1
                  = ( cons_nat @ X2 @ Xs2 ) )
               => ! [Ys2: list_nat] :
                    ( ( A22
                      = ( cons_nat @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_nat @ X2 @ Y2 )
                     => ( ~ ( ord_less_nat @ Y2 @ X2 )
                       => ~ ( ord_lexordp_nat @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_1471_lexordp_Ocases,axiom,
    ! [A1: list_int,A22: list_int] :
      ( ( ord_lexordp_int @ A1 @ A22 )
     => ( ( ( A1 = nil_int )
         => ! [Y2: int,Ys2: list_int] :
              ( A22
             != ( cons_int @ Y2 @ Ys2 ) ) )
       => ( ! [X2: int] :
              ( ? [Xs2: list_int] :
                  ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
             => ! [Y2: int] :
                  ( ? [Ys2: list_int] :
                      ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                 => ~ ( ord_less_int @ X2 @ Y2 ) ) )
         => ~ ! [X2: int,Y2: int,Xs2: list_int] :
                ( ( A1
                  = ( cons_int @ X2 @ Xs2 ) )
               => ! [Ys2: list_int] :
                    ( ( A22
                      = ( cons_int @ Y2 @ Ys2 ) )
                   => ( ~ ( ord_less_int @ X2 @ Y2 )
                     => ( ~ ( ord_less_int @ Y2 @ X2 )
                       => ~ ( ord_lexordp_int @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).

% lexordp.cases
thf(fact_1472_finite__Int,axiom,
    ! [F2: set_nat,G: set_nat] :
      ( ( ( finite_finite_nat @ F2 )
        | ( finite_finite_nat @ G ) )
     => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_1473_finite__Int,axiom,
    ! [F2: set_Pr1261947904930325089at_nat,G: set_Pr1261947904930325089at_nat] :
      ( ( ( finite6177210948735845034at_nat @ F2 )
        | ( finite6177210948735845034at_nat @ G ) )
     => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F2 @ G ) ) ) ).

% finite_Int
thf(fact_1474_ex__min__if__finite,axiom,
    ! [S: set_o] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ? [X2: $o] :
            ( ( member_o @ X2 @ S )
            & ~ ? [Xa2: $o] :
                  ( ( member_o @ Xa2 @ S )
                  & ( ord_less_o @ Xa2 @ X2 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1475_ex__min__if__finite,axiom,
    ! [S: set_assn] :
      ( ( finite_finite_assn @ S )
     => ( ( S != bot_bot_set_assn )
       => ? [X2: assn] :
            ( ( member_assn @ X2 @ S )
            & ~ ? [Xa2: assn] :
                  ( ( member_assn @ Xa2 @ S )
                  & ( ord_less_assn @ Xa2 @ X2 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1476_ex__min__if__finite,axiom,
    ! [S: set_nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat2 @ X2 @ S )
            & ~ ? [Xa2: nat] :
                  ( ( member_nat2 @ Xa2 @ S )
                  & ( ord_less_nat @ Xa2 @ X2 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1477_ex__min__if__finite,axiom,
    ! [S: set_int] :
      ( ( finite_finite_int @ S )
     => ( ( S != bot_bot_set_int )
       => ? [X2: int] :
            ( ( member_int2 @ X2 @ S )
            & ~ ? [Xa2: int] :
                  ( ( member_int2 @ Xa2 @ S )
                  & ( ord_less_int @ Xa2 @ X2 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1478_infinite__growing,axiom,
    ! [X5: set_o] :
      ( ( X5 != bot_bot_set_o )
     => ( ! [X2: $o] :
            ( ( member_o @ X2 @ X5 )
           => ? [Xa2: $o] :
                ( ( member_o @ Xa2 @ X5 )
                & ( ord_less_o @ X2 @ Xa2 ) ) )
       => ~ ( finite_finite_o @ X5 ) ) ) ).

% infinite_growing
thf(fact_1479_infinite__growing,axiom,
    ! [X5: set_nat] :
      ( ( X5 != bot_bot_set_nat )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ X5 )
           => ? [Xa2: nat] :
                ( ( member_nat2 @ Xa2 @ X5 )
                & ( ord_less_nat @ X2 @ Xa2 ) ) )
       => ~ ( finite_finite_nat @ X5 ) ) ) ).

% infinite_growing
thf(fact_1480_infinite__growing,axiom,
    ! [X5: set_int] :
      ( ( X5 != bot_bot_set_int )
     => ( ! [X2: int] :
            ( ( member_int2 @ X2 @ X5 )
           => ? [Xa2: int] :
                ( ( member_int2 @ Xa2 @ X5 )
                & ( ord_less_int @ X2 @ Xa2 ) ) )
       => ~ ( finite_finite_int @ X5 ) ) ) ).

% infinite_growing
thf(fact_1481_FI__init,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( ( fi @ nil_Pr5671120429643327159n_assn @ ( times_times_assn @ sln @ P ) @ ( times_times_assn @ sln @ Q ) @ sln @ sln @ F2 )
     => ( fI_QUERY @ P @ Q @ F2 ) ) ).

% FI_init
thf(fact_1482_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_1483_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1484_infinite__imp__nonempty,axiom,
    ! [S: set_o] :
      ( ~ ( finite_finite_o @ S )
     => ( S != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_1485_infinite__imp__nonempty,axiom,
    ! [S: set_nat] :
      ( ~ ( finite_finite_nat @ S )
     => ( S != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1486_finite__transitivity__chain,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,R: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ! [X2: produc3658429121746597890et_nat > $o] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o,Z3: produc3658429121746597890et_nat > $o] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z3 )
               => ( R @ X2 @ Z3 ) ) )
         => ( ! [X2: produc3658429121746597890et_nat > $o] :
                ( ( member6576561426505652726_nat_o @ X2 @ A3 )
               => ? [Y4: produc3658429121746597890et_nat > $o] :
                    ( ( member6576561426505652726_nat_o @ Y4 @ A3 )
                    & ( R @ X2 @ Y4 ) ) )
           => ( A3 = bot_bo7824918357723582233_nat_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1487_finite__transitivity__chain,axiom,
    ! [A3: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ! [X2: $o] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: $o,Y2: $o,Z3: $o] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z3 )
               => ( R @ X2 @ Z3 ) ) )
         => ( ! [X2: $o] :
                ( ( member_o @ X2 @ A3 )
               => ? [Y4: $o] :
                    ( ( member_o @ Y4 @ A3 )
                    & ( R @ X2 @ Y4 ) ) )
           => ( A3 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1488_finite__transitivity__chain,axiom,
    ! [A3: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ! [X2: nat] :
            ~ ( R @ X2 @ X2 )
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( ( R @ X2 @ Y2 )
             => ( ( R @ Y2 @ Z3 )
               => ( R @ X2 @ Z3 ) ) )
         => ( ! [X2: nat] :
                ( ( member_nat2 @ X2 @ A3 )
               => ? [Y4: nat] :
                    ( ( member_nat2 @ Y4 @ A3 )
                    & ( R @ X2 @ Y4 ) ) )
           => ( A3 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1489_merge__list_Oelims,axiom,
    ! [X: list_list_nat,Xa: list_list_nat,Y: list_nat] :
      ( ( ( merge_list_nat @ X @ Xa )
        = Y )
     => ( ( ( X = nil_list_nat )
         => ( ( Xa = nil_list_nat )
           => ( Y != nil_nat ) ) )
       => ( ( ( X = nil_list_nat )
           => ! [L2: list_nat] :
                ( ( Xa
                  = ( cons_list_nat @ L2 @ nil_list_nat ) )
               => ( Y != L2 ) ) )
         => ( ! [La: list_nat,Acc2: list_list_nat] :
                ( ( X
                  = ( cons_list_nat @ La @ Acc2 ) )
               => ( ( Xa = nil_list_nat )
                 => ( Y
                   != ( merge_list_nat @ nil_list_nat @ ( cons_list_nat @ La @ Acc2 ) ) ) ) )
           => ( ! [La: list_nat,Acc2: list_list_nat] :
                  ( ( X
                    = ( cons_list_nat @ La @ Acc2 ) )
                 => ! [L2: list_nat] :
                      ( ( Xa
                        = ( cons_list_nat @ L2 @ nil_list_nat ) )
                     => ( Y
                       != ( merge_list_nat @ nil_list_nat @ ( cons_list_nat @ L2 @ ( cons_list_nat @ La @ Acc2 ) ) ) ) ) )
             => ~ ! [L1: list_nat,L22: list_nat,Ls2: list_list_nat] :
                    ( ( Xa
                      = ( cons_list_nat @ L1 @ ( cons_list_nat @ L22 @ Ls2 ) ) )
                   => ( Y
                     != ( merge_list_nat @ ( cons_list_nat @ ( merge_nat @ L1 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ).

% merge_list.elims
thf(fact_1490_merge__list_Oelims,axiom,
    ! [X: list_list_int,Xa: list_list_int,Y: list_int] :
      ( ( ( merge_list_int @ X @ Xa )
        = Y )
     => ( ( ( X = nil_list_int )
         => ( ( Xa = nil_list_int )
           => ( Y != nil_int ) ) )
       => ( ( ( X = nil_list_int )
           => ! [L2: list_int] :
                ( ( Xa
                  = ( cons_list_int @ L2 @ nil_list_int ) )
               => ( Y != L2 ) ) )
         => ( ! [La: list_int,Acc2: list_list_int] :
                ( ( X
                  = ( cons_list_int @ La @ Acc2 ) )
               => ( ( Xa = nil_list_int )
                 => ( Y
                   != ( merge_list_int @ nil_list_int @ ( cons_list_int @ La @ Acc2 ) ) ) ) )
           => ( ! [La: list_int,Acc2: list_list_int] :
                  ( ( X
                    = ( cons_list_int @ La @ Acc2 ) )
                 => ! [L2: list_int] :
                      ( ( Xa
                        = ( cons_list_int @ L2 @ nil_list_int ) )
                     => ( Y
                       != ( merge_list_int @ nil_list_int @ ( cons_list_int @ L2 @ ( cons_list_int @ La @ Acc2 ) ) ) ) ) )
             => ~ ! [L1: list_int,L22: list_int,Ls2: list_list_int] :
                    ( ( Xa
                      = ( cons_list_int @ L1 @ ( cons_list_int @ L22 @ Ls2 ) ) )
                   => ( Y
                     != ( merge_list_int @ ( cons_list_int @ ( merge_int @ L1 @ L22 ) @ X ) @ Ls2 ) ) ) ) ) ) ) ) ).

% merge_list.elims
thf(fact_1491_maps__simps_I1_J,axiom,
    ! [F: nat > list_nat,X: nat,Xs: list_nat] :
      ( ( maps_nat_nat @ F @ ( cons_nat @ X @ Xs ) )
      = ( append_nat @ ( F @ X ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1492_maps__simps_I1_J,axiom,
    ! [F: nat > list_int,X: nat,Xs: list_nat] :
      ( ( maps_nat_int @ F @ ( cons_nat @ X @ Xs ) )
      = ( append_int @ ( F @ X ) @ ( maps_nat_int @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1493_maps__simps_I1_J,axiom,
    ! [F: int > list_nat,X: int,Xs: list_int] :
      ( ( maps_int_nat @ F @ ( cons_int @ X @ Xs ) )
      = ( append_nat @ ( F @ X ) @ ( maps_int_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1494_maps__simps_I1_J,axiom,
    ! [F: int > list_int,X: int,Xs: list_int] :
      ( ( maps_int_int @ F @ ( cons_int @ X @ Xs ) )
      = ( append_int @ ( F @ X ) @ ( maps_int_int @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1495_maps__simps_I1_J,axiom,
    ! [F: produc6575502325842934193n_assn > list_nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( maps_P5986043149438648064sn_nat @ F @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( append_nat @ ( F @ X ) @ ( maps_P5986043149438648064sn_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1496_maps__simps_I1_J,axiom,
    ! [F: produc6575502325842934193n_assn > list_int,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( maps_P5983552678929597788sn_int @ F @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( append_int @ ( F @ X ) @ ( maps_P5983552678929597788sn_int @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1497_rotate1_Osimps_I2_J,axiom,
    ! [X: a,Xs: list_a] :
      ( ( rotate1_a @ ( cons_a @ X @ Xs ) )
      = ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).

% rotate1.simps(2)
thf(fact_1498_rotate1_Osimps_I2_J,axiom,
    ! [X: b,Xs: list_b] :
      ( ( rotate1_b @ ( cons_b @ X @ Xs ) )
      = ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) ) ).

% rotate1.simps(2)
thf(fact_1499_rotate1_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( rotate1_nat @ ( cons_nat @ X @ Xs ) )
      = ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) ) ).

% rotate1.simps(2)
thf(fact_1500_rotate1_Osimps_I2_J,axiom,
    ! [X: int,Xs: list_int] :
      ( ( rotate1_int @ ( cons_int @ X @ Xs ) )
      = ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) ).

% rotate1.simps(2)
thf(fact_1501_rotate1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( rotate328796349445179396n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ) ).

% rotate1.simps(2)
thf(fact_1502_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys: list_a,Zs3: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_a )
         => ( ( Ys = nil_a )
            & ( Zs3 = nil_a ) ) )
        & ( ( Xss2 != nil_list_a )
         => ? [Xss1: list_list_a,Xs3: list_a,Xs5: list_a,Xss22: list_list_a] :
              ( ( Xss2
                = ( append_list_a @ Xss1 @ ( cons_list_a @ ( append_a @ Xs3 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_a @ ( concat_a @ Xss1 ) @ Xs3 ) )
              & ( Zs3
                = ( append_a @ Xs5 @ ( concat_a @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1503_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_b,Ys: list_b,Zs3: list_b] :
      ( ( ( concat_b @ Xss2 )
        = ( append_b @ Ys @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_b )
         => ( ( Ys = nil_b )
            & ( Zs3 = nil_b ) ) )
        & ( ( Xss2 != nil_list_b )
         => ? [Xss1: list_list_b,Xs3: list_b,Xs5: list_b,Xss22: list_list_b] :
              ( ( Xss2
                = ( append_list_b @ Xss1 @ ( cons_list_b @ ( append_b @ Xs3 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_b @ ( concat_b @ Xss1 ) @ Xs3 ) )
              & ( Zs3
                = ( append_b @ Xs5 @ ( concat_b @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1504_concat__eq__append__conv,axiom,
    ! [Xss2: list_l6351802567095793725n_assn,Ys: list_P8527749157015355191n_assn,Zs3: list_P8527749157015355191n_assn] :
      ( ( ( concat6144485081897559666n_assn @ Xss2 )
        = ( append282499809098378956n_assn @ Ys @ Zs3 ) )
      = ( ( ( Xss2 = nil_li5476096274760905021n_assn )
         => ( ( Ys = nil_Pr5671120429643327159n_assn )
            & ( Zs3 = nil_Pr5671120429643327159n_assn ) ) )
        & ( ( Xss2 != nil_li5476096274760905021n_assn )
         => ? [Xss1: list_l6351802567095793725n_assn,Xs3: list_P8527749157015355191n_assn,Xs5: list_P8527749157015355191n_assn,Xss22: list_l6351802567095793725n_assn] :
              ( ( Xss2
                = ( append2733831349823859410n_assn @ Xss1 @ ( cons_l2423627976422276333n_assn @ ( append282499809098378956n_assn @ Xs3 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append282499809098378956n_assn @ ( concat6144485081897559666n_assn @ Xss1 ) @ Xs3 ) )
              & ( Zs3
                = ( append282499809098378956n_assn @ Xs5 @ ( concat6144485081897559666n_assn @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1505_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys = nil_nat )
            & ( Zs3 = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs3: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
              & ( Zs3
                = ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1506_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( concat_int @ Xss2 )
        = ( append_int @ Ys @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_int )
         => ( ( Ys = nil_int )
            & ( Zs3 = nil_int ) ) )
        & ( ( Xss2 != nil_list_int )
         => ? [Xss1: list_list_int,Xs3: list_int,Xs5: list_int,Xss22: list_list_int] :
              ( ( Xss2
                = ( append_list_int @ Xss1 @ ( cons_list_int @ ( append_int @ Xs3 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_int @ ( concat_int @ Xss1 ) @ Xs3 ) )
              & ( Zs3
                = ( append_int @ Xs5 @ ( concat_int @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1507_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( rotate1_a @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% rotate1_is_Nil_conv
thf(fact_1508_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_b] :
      ( ( ( rotate1_b @ Xs )
        = nil_b )
      = ( Xs = nil_b ) ) ).

% rotate1_is_Nil_conv
thf(fact_1509_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( rotate328796349445179396n_assn @ Xs )
        = nil_Pr5671120429643327159n_assn )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% rotate1_is_Nil_conv
thf(fact_1510_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( rotate1_nat @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_1511_rotate1__is__Nil__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( rotate1_int @ Xs )
        = nil_int )
      = ( Xs = nil_int ) ) ).

% rotate1_is_Nil_conv
thf(fact_1512_concat__append,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
      = ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).

% concat_append
thf(fact_1513_concat__append,axiom,
    ! [Xs: list_list_int,Ys: list_list_int] :
      ( ( concat_int @ ( append_list_int @ Xs @ Ys ) )
      = ( append_int @ ( concat_int @ Xs ) @ ( concat_int @ Ys ) ) ) ).

% concat_append
thf(fact_1514_assn__aci_I8_J,axiom,
    ! [X: assn,Y: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ X @ Y ) )
      = ( sup_sup_assn @ X @ Y ) ) ).

% assn_aci(8)
thf(fact_1515_assn__aci_I7_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) )
      = ( sup_sup_assn @ Y @ ( sup_sup_assn @ X @ Z ) ) ) ).

% assn_aci(7)
thf(fact_1516_assn__aci_I5_J,axiom,
    ( sup_sup_assn
    = ( ^ [X3: assn,Y3: assn] : ( sup_sup_assn @ Y3 @ X3 ) ) ) ).

% assn_aci(5)
thf(fact_1517_norm__assertion__simps_I32_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ X )
      = X ) ).

% norm_assertion_simps(32)
thf(fact_1518_norm__assertion__simps_I15_J,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( sup_sup_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( sup_sup_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% norm_assertion_simps(15)
thf(fact_1519_map__concat,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_1520_map__concat,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_l6351802567095793725n_assn] :
      ( ( map_Pr8991440229025900053n_assn @ F @ ( concat6144485081897559666n_assn @ Xs ) )
      = ( concat_assn @ ( map_li562537959812877739t_assn @ ( map_Pr8991440229025900053n_assn @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_1521_List_Obind__def,axiom,
    ( bind_a3542047475819770682n_assn
    = ( ^ [Xs3: list_a,F3: a > list_P8527749157015355191n_assn] : ( concat6144485081897559666n_assn @ ( map_a_7516433405034028541n_assn @ F3 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_1522_List_Obind__def,axiom,
    ( bind_a_nat
    = ( ^ [Xs3: list_a,F3: a > list_nat] : ( concat_nat @ ( map_a_list_nat @ F3 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_1523_List_Obind__def,axiom,
    ( bind_a_int
    = ( ^ [Xs3: list_a,F3: a > list_int] : ( concat_int @ ( map_a_list_int @ F3 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_1524_List_Obind__def,axiom,
    ( bind_a_b
    = ( ^ [Xs3: list_a,F3: a > list_b] : ( concat_b @ ( map_a_list_b @ F3 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_1525_List_Obind__def,axiom,
    ( bind_a_a
    = ( ^ [Xs3: list_a,F3: a > list_a] : ( concat_a @ ( map_a_list_a @ F3 @ Xs3 ) ) ) ) ).

% List.bind_def
thf(fact_1526_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_a @ nil_a )
    = nil_a ) ).

% rotate1.simps(1)
thf(fact_1527_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_b @ nil_b )
    = nil_b ) ).

% rotate1.simps(1)
thf(fact_1528_rotate1_Osimps_I1_J,axiom,
    ( ( rotate328796349445179396n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% rotate1.simps(1)
thf(fact_1529_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_1530_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_int @ nil_int )
    = nil_int ) ).

% rotate1.simps(1)
thf(fact_1531_rotate1__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( rotate1_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( rotate1_nat @ Xs ) ) ) ).

% rotate1_map
thf(fact_1532_rotate1__map,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( rotate1_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) )
      = ( map_Pr8991440229025900053n_assn @ F @ ( rotate328796349445179396n_assn @ Xs ) ) ) ).

% rotate1_map
thf(fact_1533_concat_Osimps_I1_J,axiom,
    ( ( concat_a @ nil_list_a )
    = nil_a ) ).

% concat.simps(1)
thf(fact_1534_concat_Osimps_I1_J,axiom,
    ( ( concat_b @ nil_list_b )
    = nil_b ) ).

% concat.simps(1)
thf(fact_1535_concat_Osimps_I1_J,axiom,
    ( ( concat6144485081897559666n_assn @ nil_li5476096274760905021n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% concat.simps(1)
thf(fact_1536_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_1537_concat_Osimps_I1_J,axiom,
    ( ( concat_int @ nil_list_int )
    = nil_int ) ).

% concat.simps(1)
thf(fact_1538_concat_Osimps_I2_J,axiom,
    ! [X: list_nat,Xs: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X @ Xs ) )
      = ( append_nat @ X @ ( concat_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_1539_concat_Osimps_I2_J,axiom,
    ! [X: list_int,Xs: list_list_int] :
      ( ( concat_int @ ( cons_list_int @ X @ Xs ) )
      = ( append_int @ X @ ( concat_int @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_1540_maps__simps_I2_J,axiom,
    ! [F: a > list_a] :
      ( ( maps_a_a @ F @ nil_a )
      = nil_a ) ).

% maps_simps(2)
thf(fact_1541_maps__simps_I2_J,axiom,
    ! [F: a > list_b] :
      ( ( maps_a_b @ F @ nil_a )
      = nil_b ) ).

% maps_simps(2)
thf(fact_1542_maps__simps_I2_J,axiom,
    ! [F: a > list_nat] :
      ( ( maps_a_nat @ F @ nil_a )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1543_maps__simps_I2_J,axiom,
    ! [F: a > list_int] :
      ( ( maps_a_int @ F @ nil_a )
      = nil_int ) ).

% maps_simps(2)
thf(fact_1544_maps__simps_I2_J,axiom,
    ! [F: b > list_a] :
      ( ( maps_b_a @ F @ nil_b )
      = nil_a ) ).

% maps_simps(2)
thf(fact_1545_maps__simps_I2_J,axiom,
    ! [F: b > list_b] :
      ( ( maps_b_b @ F @ nil_b )
      = nil_b ) ).

% maps_simps(2)
thf(fact_1546_maps__simps_I2_J,axiom,
    ! [F: b > list_nat] :
      ( ( maps_b_nat @ F @ nil_b )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1547_maps__simps_I2_J,axiom,
    ! [F: b > list_int] :
      ( ( maps_b_int @ F @ nil_b )
      = nil_int ) ).

% maps_simps(2)
thf(fact_1548_maps__simps_I2_J,axiom,
    ! [F: nat > list_a] :
      ( ( maps_nat_a @ F @ nil_nat )
      = nil_a ) ).

% maps_simps(2)
thf(fact_1549_maps__simps_I2_J,axiom,
    ! [F: nat > list_b] :
      ( ( maps_nat_b @ F @ nil_nat )
      = nil_b ) ).

% maps_simps(2)
thf(fact_1550_merge__list_Osimps_I1_J,axiom,
    ( ( merge_list_nat @ nil_list_nat @ nil_list_nat )
    = nil_nat ) ).

% merge_list.simps(1)
thf(fact_1551_merge__list_Osimps_I1_J,axiom,
    ( ( merge_list_int @ nil_list_int @ nil_list_int )
    = nil_int ) ).

% merge_list.simps(1)
thf(fact_1552_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys @ Zs3 ) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs2: list_nat,Xs4: list_nat,Xss23: list_list_nat] :
            ( ( Xss2
              = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs4 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
            & ( Zs3
              = ( append_nat @ Xs4 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1553_concat__eq__appendD,axiom,
    ! [Xss2: list_list_int,Ys: list_int,Zs3: list_int] :
      ( ( ( concat_int @ Xss2 )
        = ( append_int @ Ys @ Zs3 ) )
     => ( ( Xss2 != nil_list_int )
       => ? [Xss12: list_list_int,Xs2: list_int,Xs4: list_int,Xss23: list_list_int] :
            ( ( Xss2
              = ( append_list_int @ Xss12 @ ( cons_list_int @ ( append_int @ Xs2 @ Xs4 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_int @ ( concat_int @ Xss12 ) @ Xs2 ) )
            & ( Zs3
              = ( append_int @ Xs4 @ ( concat_int @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1554_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > assn] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X6: $o] :
              ( ( member_o @ X6 @ S )
              & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic7708394068118247271o_assn @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1555_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > assn] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat2 @ X6 @ S )
              & ( ord_less_assn @ ( F @ X6 ) @ ( F @ ( lattic2187264146484958483t_assn @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1556_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X6: $o] :
              ( ( member_o @ X6 @ S )
              & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1557_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat2 @ X6 @ S )
              & ( ord_less_nat @ ( F @ X6 ) @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1558_arg__min__if__finite_I2_J,axiom,
    ! [S: set_o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ~ ? [X6: $o] :
              ( ( member_o @ X6 @ S )
              & ( ord_less_int @ ( F @ X6 ) @ ( F @ ( lattic2773365557947402859_o_int @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1559_arg__min__if__finite_I2_J,axiom,
    ! [S: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ~ ? [X6: nat] :
              ( ( member_nat2 @ X6 @ S )
              & ( ord_less_int @ ( F @ X6 ) @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1560_Sup__fin_Ounion,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B3 )
         => ( ( B3 != bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ ( sup_sup_set_assn @ A3 @ B3 ) )
              = ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ ( lattic2150320897289308081n_assn @ B3 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1561_Sup__fin_Ounion,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B3 )
         => ( ( B3 != bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
              = ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ ( lattic3835124923745554447et_nat @ B3 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1562_Sup__fin_Ounion,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ( B3 != bot_bot_set_o )
           => ( ( lattic1508158080041050831_fin_o @ ( sup_sup_set_o @ A3 @ B3 ) )
              = ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ ( lattic1508158080041050831_fin_o @ B3 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1563_Sup__fin_Ounion,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
              = ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B3 ) ) ) ) ) ) ) ).

% Sup_fin.union
thf(fact_1564_Inf__fin_Ounion,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B3 )
         => ( ( B3 != bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ ( sup_sup_set_assn @ A3 @ B3 ) )
              = ( inf_inf_assn @ ( lattic47131356835913163n_assn @ A3 ) @ ( lattic47131356835913163n_assn @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1565_Inf__fin_Ounion,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B3 )
         => ( ( B3 != bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ ( sup_sup_set_set_nat @ A3 @ B3 ) )
              = ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ A3 ) @ ( lattic3014633134055518761et_nat @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1566_Inf__fin_Ounion,axiom,
    ! [A3: set_Product_unit,B3: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( A3 != bot_bo3957492148770167129t_unit )
       => ( ( finite4290736615968046902t_unit @ B3 )
         => ( ( B3 != bot_bo3957492148770167129t_unit )
           => ( ( lattic1263872656861969706t_unit @ ( sup_su793286257634532545t_unit @ A3 @ B3 ) )
              = ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ A3 ) @ ( lattic1263872656861969706t_unit @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1567_Inf__fin_Ounion,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( A3 != bot_bo3083307316010499117at_nat )
       => ( ( finite9047747110432174090at_nat @ B3 )
         => ( ( B3 != bot_bo3083307316010499117at_nat )
           => ( ( lattic30941717366863870at_nat @ ( sup_su3642409539654194069at_nat @ A3 @ B3 ) )
              = ( inf_in2572325071724192079at_nat @ ( lattic30941717366863870at_nat @ A3 ) @ ( lattic30941717366863870at_nat @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1568_Inf__fin_Ounion,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ( B3 != bot_bot_set_o )
           => ( ( lattic4107685809792843317_fin_o @ ( sup_sup_set_o @ A3 @ B3 ) )
              = ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ A3 ) @ ( lattic4107685809792843317_fin_o @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1569_Inf__fin_Ounion,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
              = ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic5238388535129920115in_nat @ B3 ) ) ) ) ) ) ) ).

% Inf_fin.union
thf(fact_1570_product_Osimps_I1_J,axiom,
    ! [Uu: list_assn] :
      ( ( product_assn_assn @ nil_assn @ Uu )
      = nil_Pr5671120429643327159n_assn ) ).

% product.simps(1)
thf(fact_1571_butlast__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1572_butlast__snoc,axiom,
    ! [Xs: list_b,X: b] :
      ( ( butlast_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1573_butlast__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( butlast_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1574_butlast__snoc,axiom,
    ! [Xs: list_int,X: int] :
      ( ( butlast_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1575_butlast__snoc,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( butlas3012047794866324995n_assn @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1576_last__snoc,axiom,
    ! [Xs: list_a,X: a] :
      ( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = X ) ).

% last_snoc
thf(fact_1577_last__snoc,axiom,
    ! [Xs: list_b,X: b] :
      ( ( last_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
      = X ) ).

% last_snoc
thf(fact_1578_last__snoc,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% last_snoc
thf(fact_1579_last__snoc,axiom,
    ! [Xs: list_int,X: int] :
      ( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = X ) ).

% last_snoc
thf(fact_1580_last__snoc,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( last_P8723976779861936080n_assn @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) )
      = X ) ).

% last_snoc
thf(fact_1581_Max__less__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_o @ X3 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_1582_Max__less__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_1583_Max__less__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_int @ X3 @ X ) ) ) ) ) ) ).

% Max_less_iff
thf(fact_1584_concat__conv__foldr,axiom,
    ( concat_a
    = ( ^ [Xss3: list_list_a] : ( foldr_list_a_list_a @ append_a @ Xss3 @ nil_a ) ) ) ).

% concat_conv_foldr
thf(fact_1585_concat__conv__foldr,axiom,
    ( concat_b
    = ( ^ [Xss3: list_list_b] : ( foldr_list_b_list_b @ append_b @ Xss3 @ nil_b ) ) ) ).

% concat_conv_foldr
thf(fact_1586_concat__conv__foldr,axiom,
    ( concat6144485081897559666n_assn
    = ( ^ [Xss3: list_l6351802567095793725n_assn] : ( foldr_4640886882926046823n_assn @ append282499809098378956n_assn @ Xss3 @ nil_Pr5671120429643327159n_assn ) ) ) ).

% concat_conv_foldr
thf(fact_1587_concat__conv__foldr,axiom,
    ( concat_nat
    = ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).

% concat_conv_foldr
thf(fact_1588_concat__conv__foldr,axiom,
    ( concat_int
    = ( ^ [Xss3: list_list_int] : ( foldr_4541567342299342241st_int @ append_int @ Xss3 @ nil_int ) ) ) ).

% concat_conv_foldr
thf(fact_1589_Min__gr__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_o @ X @ X3 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1590_Min__gr__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1591_Min__gr__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ X @ ( lattic8718645017227715691in_int @ A3 ) )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_int @ X @ X3 ) ) ) ) ) ) ).

% Min_gr_iff
thf(fact_1592_SuccI,axiom,
    ! [Kl: list_P7985473006766602707_nat_o,K: produc3658429121746597890et_nat > $o,Kl2: set_li630567559872716595_nat_o] :
      ( ( member4792007445727162748_nat_o @ ( append6126940767462283112_nat_o @ Kl @ ( cons_P6219271836124797827_nat_o @ K @ nil_Pr1626566771615838163_nat_o ) ) @ Kl2 )
     => ( member6576561426505652726_nat_o @ K @ ( bNF_Gr992270151760408296_nat_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1593_SuccI,axiom,
    ! [Kl: list_a,K: a,Kl2: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
     => ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1594_SuccI,axiom,
    ! [Kl: list_b,K: b,Kl2: set_list_b] :
      ( ( member_list_b @ ( append_b @ Kl @ ( cons_b @ K @ nil_b ) ) @ Kl2 )
     => ( member_b2 @ K @ ( bNF_Greatest_Succ_b @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1595_SuccI,axiom,
    ! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
     => ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1596_SuccI,axiom,
    ! [Kl: list_int,K: int,Kl2: set_list_int] :
      ( ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 )
     => ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1597_SuccI,axiom,
    ! [Kl: list_P8527749157015355191n_assn,K: produc6575502325842934193n_assn,Kl2: set_li5131720305576846103n_assn] :
      ( ( member852475432509897056n_assn @ ( append282499809098378956n_assn @ Kl @ ( cons_P2971678138204555879n_assn @ K @ nil_Pr5671120429643327159n_assn ) ) @ Kl2 )
     => ( member7957490590177025114n_assn @ K @ ( bNF_Gr3216292445876487756n_assn @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1598_foldr__append,axiom,
    ! [F: assn > assn > assn,Xs: list_assn,Ys: list_assn,A: assn] :
      ( ( foldr_assn_assn @ F @ ( append_assn @ Xs @ Ys ) @ A )
      = ( foldr_assn_assn @ F @ Xs @ ( foldr_assn_assn @ F @ Ys @ A ) ) ) ).

% foldr_append
thf(fact_1599_last__appendL,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys = nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys ) )
        = ( last_a @ Xs ) ) ) ).

% last_appendL
thf(fact_1600_last__appendL,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( Ys = nil_b )
     => ( ( last_b @ ( append_b @ Xs @ Ys ) )
        = ( last_b @ Xs ) ) ) ).

% last_appendL
thf(fact_1601_last__appendL,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Ys = nil_Pr5671120429643327159n_assn )
     => ( ( last_P8723976779861936080n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
        = ( last_P8723976779861936080n_assn @ Xs ) ) ) ).

% last_appendL
thf(fact_1602_last__appendL,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Xs ) ) ) ).

% last_appendL
thf(fact_1603_last__appendL,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys = nil_int )
     => ( ( last_int @ ( append_int @ Xs @ Ys ) )
        = ( last_int @ Xs ) ) ) ).

% last_appendL
thf(fact_1604_last__appendR,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( Ys != nil_a )
     => ( ( last_a @ ( append_a @ Xs @ Ys ) )
        = ( last_a @ Ys ) ) ) ).

% last_appendR
thf(fact_1605_last__appendR,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( Ys != nil_b )
     => ( ( last_b @ ( append_b @ Xs @ Ys ) )
        = ( last_b @ Ys ) ) ) ).

% last_appendR
thf(fact_1606_last__appendR,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( Ys != nil_Pr5671120429643327159n_assn )
     => ( ( last_P8723976779861936080n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
        = ( last_P8723976779861936080n_assn @ Ys ) ) ) ).

% last_appendR
thf(fact_1607_last__appendR,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Ys ) ) ) ).

% last_appendR
thf(fact_1608_last__appendR,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( Ys != nil_int )
     => ( ( last_int @ ( append_int @ Xs @ Ys ) )
        = ( last_int @ Ys ) ) ) ).

% last_appendR
thf(fact_1609_inf__Sup__absorb,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ A @ A3 )
       => ( ( inf_in1318976480646536635_nat_o @ A @ ( lattic7320199455484906628_nat_o @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1610_inf__Sup__absorb,axiom,
    ! [A3: set_assn,A: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( member_assn @ A @ A3 )
       => ( ( inf_inf_assn @ A @ ( lattic2150320897289308081n_assn @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1611_inf__Sup__absorb,axiom,
    ! [A3: set_nat,A: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ A @ A3 )
       => ( ( inf_inf_nat @ A @ ( lattic1093996805478795353in_nat @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1612_inf__Sup__absorb,axiom,
    ! [A3: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ A @ A3 )
       => ( ( inf_inf_set_nat @ A @ ( lattic3835124923745554447et_nat @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1613_inf__Sup__absorb,axiom,
    ! [A3: set_Product_unit,A: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( member_Product_unit @ A @ A3 )
       => ( ( inf_inf_Product_unit @ A @ ( lattic5294303975357428420t_unit @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1614_inf__Sup__absorb,axiom,
    ! [A3: set_se7855581050983116737at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( member2643936169264416010at_nat @ A @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A @ ( lattic1541023418247406232at_nat @ A3 ) )
          = A ) ) ) ).

% inf_Sup_absorb
thf(fact_1615_append__butlast__last__id,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1616_append__butlast__last__id,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ( ( append_b @ ( butlast_b @ Xs ) @ ( cons_b @ ( last_b @ Xs ) @ nil_b ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1617_append__butlast__last__id,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1618_append__butlast__last__id,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ( ( append_int @ ( butlast_int @ Xs ) @ ( cons_int @ ( last_int @ Xs ) @ nil_int ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1619_append__butlast__last__id,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ( append282499809098378956n_assn @ ( butlas3012047794866324995n_assn @ Xs ) @ ( cons_P2971678138204555879n_assn @ ( last_P8723976779861936080n_assn @ Xs ) @ nil_Pr5671120429643327159n_assn ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1620_butlast_Osimps_I1_J,axiom,
    ( ( butlast_a @ nil_a )
    = nil_a ) ).

% butlast.simps(1)
thf(fact_1621_butlast_Osimps_I1_J,axiom,
    ( ( butlast_b @ nil_b )
    = nil_b ) ).

% butlast.simps(1)
thf(fact_1622_butlast_Osimps_I1_J,axiom,
    ( ( butlas3012047794866324995n_assn @ nil_Pr5671120429643327159n_assn )
    = nil_Pr5671120429643327159n_assn ) ).

% butlast.simps(1)
thf(fact_1623_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_1624_butlast_Osimps_I1_J,axiom,
    ( ( butlast_int @ nil_int )
    = nil_int ) ).

% butlast.simps(1)
thf(fact_1625_map__butlast,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( map_nat_nat @ F @ ( butlast_nat @ Xs ) )
      = ( butlast_nat @ ( map_nat_nat @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_1626_map__butlast,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( map_Pr8991440229025900053n_assn @ F @ ( butlas3012047794866324995n_assn @ Xs ) )
      = ( butlast_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) ) ) ).

% map_butlast
thf(fact_1627_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_a,X: a,Ys: list_a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
        = Ys )
      = ( ( Ys != nil_a )
        & ( ( butlast_a @ Ys )
          = Xs )
        & ( ( last_a @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1628_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_b,X: b,Ys: list_b] :
      ( ( ( append_b @ Xs @ ( cons_b @ X @ nil_b ) )
        = Ys )
      = ( ( Ys != nil_b )
        & ( ( butlast_b @ Ys )
          = Xs )
        & ( ( last_b @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1629_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = Ys )
      = ( ( Ys != nil_nat )
        & ( ( butlast_nat @ Ys )
          = Xs )
        & ( ( last_nat @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1630_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] :
      ( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
        = Ys )
      = ( ( Ys != nil_int )
        & ( ( butlast_int @ Ys )
          = Xs )
        & ( ( last_int @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1631_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) )
        = Ys )
      = ( ( Ys != nil_Pr5671120429643327159n_assn )
        & ( ( butlas3012047794866324995n_assn @ Ys )
          = Xs )
        & ( ( last_P8723976779861936080n_assn @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1632_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_a,Xs: list_a,X: a] :
      ( ( Ys
        = ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = ( ( Ys != nil_a )
        & ( ( butlast_a @ Ys )
          = Xs )
        & ( ( last_a @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast'
thf(fact_1633_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_b,Xs: list_b,X: b] :
      ( ( Ys
        = ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
      = ( ( Ys != nil_b )
        & ( ( butlast_b @ Ys )
          = Xs )
        & ( ( last_b @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast'
thf(fact_1634_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_nat,Xs: list_nat,X: nat] :
      ( ( Ys
        = ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = ( ( Ys != nil_nat )
        & ( ( butlast_nat @ Ys )
          = Xs )
        & ( ( last_nat @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast'
thf(fact_1635_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_int,Xs: list_int,X: int] :
      ( ( Ys
        = ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = ( ( Ys != nil_int )
        & ( ( butlast_int @ Ys )
          = Xs )
        & ( ( last_int @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast'
thf(fact_1636_snoc__eq__iff__butlast_H,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Ys
        = ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) )
      = ( ( Ys != nil_Pr5671120429643327159n_assn )
        & ( ( butlas3012047794866324995n_assn @ Ys )
          = Xs )
        & ( ( last_P8723976779861936080n_assn @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast'
thf(fact_1637_Min__in,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( member_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ A3 ) ) ) ).

% Min_in
thf(fact_1638_Min__in,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat2 @ ( lattic8721135487736765967in_nat @ A3 ) @ A3 ) ) ) ).

% Min_in
thf(fact_1639_Max__in,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( member_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_1640_Max__in,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A3 ) @ A3 ) ) ) ).

% Max_in
thf(fact_1641_last_Osimps,axiom,
    ! [Xs: list_a,X: a] :
      ( ( ( Xs = nil_a )
       => ( ( last_a @ ( cons_a @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_a )
       => ( ( last_a @ ( cons_a @ X @ Xs ) )
          = ( last_a @ Xs ) ) ) ) ).

% last.simps
thf(fact_1642_last_Osimps,axiom,
    ! [Xs: list_b,X: b] :
      ( ( ( Xs = nil_b )
       => ( ( last_b @ ( cons_b @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_b )
       => ( ( last_b @ ( cons_b @ X @ Xs ) )
          = ( last_b @ Xs ) ) ) ) ).

% last.simps
thf(fact_1643_last_Osimps,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
          = ( last_nat @ Xs ) ) ) ) ).

% last.simps
thf(fact_1644_last_Osimps,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( Xs = nil_int )
       => ( ( last_int @ ( cons_int @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_int )
       => ( ( last_int @ ( cons_int @ X @ Xs ) )
          = ( last_int @ Xs ) ) ) ) ).

% last.simps
thf(fact_1645_last_Osimps,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ( Xs = nil_Pr5671120429643327159n_assn )
       => ( ( last_P8723976779861936080n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = X ) )
      & ( ( Xs != nil_Pr5671120429643327159n_assn )
       => ( ( last_P8723976779861936080n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( last_P8723976779861936080n_assn @ Xs ) ) ) ) ).

% last.simps
thf(fact_1646_last__ConsL,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs = nil_a )
     => ( ( last_a @ ( cons_a @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1647_last__ConsL,axiom,
    ! [Xs: list_b,X: b] :
      ( ( Xs = nil_b )
     => ( ( last_b @ ( cons_b @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1648_last__ConsL,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1649_last__ConsL,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs = nil_int )
     => ( ( last_int @ ( cons_int @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1650_last__ConsL,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Xs = nil_Pr5671120429643327159n_assn )
     => ( ( last_P8723976779861936080n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
        = X ) ) ).

% last_ConsL
thf(fact_1651_last__ConsR,axiom,
    ! [Xs: list_a,X: a] :
      ( ( Xs != nil_a )
     => ( ( last_a @ ( cons_a @ X @ Xs ) )
        = ( last_a @ Xs ) ) ) ).

% last_ConsR
thf(fact_1652_last__ConsR,axiom,
    ! [Xs: list_b,X: b] :
      ( ( Xs != nil_b )
     => ( ( last_b @ ( cons_b @ X @ Xs ) )
        = ( last_b @ Xs ) ) ) ).

% last_ConsR
thf(fact_1653_last__ConsR,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X @ Xs ) )
        = ( last_nat @ Xs ) ) ) ).

% last_ConsR
thf(fact_1654_last__ConsR,axiom,
    ! [Xs: list_int,X: int] :
      ( ( Xs != nil_int )
     => ( ( last_int @ ( cons_int @ X @ Xs ) )
        = ( last_int @ Xs ) ) ) ).

% last_ConsR
thf(fact_1655_last__ConsR,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ( last_P8723976779861936080n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
        = ( last_P8723976779861936080n_assn @ Xs ) ) ) ).

% last_ConsR
thf(fact_1656_last__map,axiom,
    ! [Xs: list_a,F: a > nat] :
      ( ( Xs != nil_a )
     => ( ( last_nat @ ( map_a_nat @ F @ Xs ) )
        = ( F @ ( last_a @ Xs ) ) ) ) ).

% last_map
thf(fact_1657_last__map,axiom,
    ! [Xs: list_b,F: b > nat] :
      ( ( Xs != nil_b )
     => ( ( last_nat @ ( map_b_nat @ F @ Xs ) )
        = ( F @ ( last_b @ Xs ) ) ) ) ).

% last_map
thf(fact_1658_last__map,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > nat] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ( last_nat @ ( map_Pr7570552894071451325sn_nat @ F @ Xs ) )
        = ( F @ ( last_P8723976779861936080n_assn @ Xs ) ) ) ) ).

% last_map
thf(fact_1659_last__map,axiom,
    ! [Xs: list_int,F: int > nat] :
      ( ( Xs != nil_int )
     => ( ( last_nat @ ( map_int_nat @ F @ Xs ) )
        = ( F @ ( last_int @ Xs ) ) ) ) ).

% last_map
thf(fact_1660_last__map,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_1661_last__map,axiom,
    ! [Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ( last_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) )
        = ( F @ ( last_P8723976779861936080n_assn @ Xs ) ) ) ) ).

% last_map
thf(fact_1662_last__append,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( ( Ys = nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys ) )
          = ( last_a @ Xs ) ) )
      & ( ( Ys != nil_a )
       => ( ( last_a @ ( append_a @ Xs @ Ys ) )
          = ( last_a @ Ys ) ) ) ) ).

% last_append
thf(fact_1663_last__append,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( ( Ys = nil_b )
       => ( ( last_b @ ( append_b @ Xs @ Ys ) )
          = ( last_b @ Xs ) ) )
      & ( ( Ys != nil_b )
       => ( ( last_b @ ( append_b @ Xs @ Ys ) )
          = ( last_b @ Ys ) ) ) ) ).

% last_append
thf(fact_1664_last__append,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( Ys = nil_Pr5671120429643327159n_assn )
       => ( ( last_P8723976779861936080n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
          = ( last_P8723976779861936080n_assn @ Xs ) ) )
      & ( ( Ys != nil_Pr5671120429643327159n_assn )
       => ( ( last_P8723976779861936080n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
          = ( last_P8723976779861936080n_assn @ Ys ) ) ) ) ).

% last_append
thf(fact_1665_last__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Ys ) ) ) ) ).

% last_append
thf(fact_1666_last__append,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( ( Ys = nil_int )
       => ( ( last_int @ ( append_int @ Xs @ Ys ) )
          = ( last_int @ Xs ) ) )
      & ( ( Ys != nil_int )
       => ( ( last_int @ ( append_int @ Xs @ Ys ) )
          = ( last_int @ Ys ) ) ) ) ).

% last_append
thf(fact_1667_longest__common__suffix,axiom,
    ! [Xs: list_a,Ys: list_a] :
    ? [Ss: list_a,Xs4: list_a,Ys4: list_a] :
      ( ( Xs
        = ( append_a @ Xs4 @ Ss ) )
      & ( Ys
        = ( append_a @ Ys4 @ Ss ) )
      & ( ( Xs4 = nil_a )
        | ( Ys4 = nil_a )
        | ( ( last_a @ Xs4 )
         != ( last_a @ Ys4 ) ) ) ) ).

% longest_common_suffix
thf(fact_1668_longest__common__suffix,axiom,
    ! [Xs: list_b,Ys: list_b] :
    ? [Ss: list_b,Xs4: list_b,Ys4: list_b] :
      ( ( Xs
        = ( append_b @ Xs4 @ Ss ) )
      & ( Ys
        = ( append_b @ Ys4 @ Ss ) )
      & ( ( Xs4 = nil_b )
        | ( Ys4 = nil_b )
        | ( ( last_b @ Xs4 )
         != ( last_b @ Ys4 ) ) ) ) ).

% longest_common_suffix
thf(fact_1669_longest__common__suffix,axiom,
    ! [Xs: list_P8527749157015355191n_assn,Ys: list_P8527749157015355191n_assn] :
    ? [Ss: list_P8527749157015355191n_assn,Xs4: list_P8527749157015355191n_assn,Ys4: list_P8527749157015355191n_assn] :
      ( ( Xs
        = ( append282499809098378956n_assn @ Xs4 @ Ss ) )
      & ( Ys
        = ( append282499809098378956n_assn @ Ys4 @ Ss ) )
      & ( ( Xs4 = nil_Pr5671120429643327159n_assn )
        | ( Ys4 = nil_Pr5671120429643327159n_assn )
        | ( ( last_P8723976779861936080n_assn @ Xs4 )
         != ( last_P8723976779861936080n_assn @ Ys4 ) ) ) ) ).

% longest_common_suffix
thf(fact_1670_longest__common__suffix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ss: list_nat,Xs4: list_nat,Ys4: list_nat] :
      ( ( Xs
        = ( append_nat @ Xs4 @ Ss ) )
      & ( Ys
        = ( append_nat @ Ys4 @ Ss ) )
      & ( ( Xs4 = nil_nat )
        | ( Ys4 = nil_nat )
        | ( ( last_nat @ Xs4 )
         != ( last_nat @ Ys4 ) ) ) ) ).

% longest_common_suffix
thf(fact_1671_longest__common__suffix,axiom,
    ! [Xs: list_int,Ys: list_int] :
    ? [Ss: list_int,Xs4: list_int,Ys4: list_int] :
      ( ( Xs
        = ( append_int @ Xs4 @ Ss ) )
      & ( Ys
        = ( append_int @ Ys4 @ Ss ) )
      & ( ( Xs4 = nil_int )
        | ( Ys4 = nil_int )
        | ( ( last_int @ Xs4 )
         != ( last_int @ Ys4 ) ) ) ) ).

% longest_common_suffix
thf(fact_1672_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_a,X: a] :
      ( ( ( Xs = nil_a )
       => ( ( butlast_a @ ( cons_a @ X @ Xs ) )
          = nil_a ) )
      & ( ( Xs != nil_a )
       => ( ( butlast_a @ ( cons_a @ X @ Xs ) )
          = ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1673_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_b,X: b] :
      ( ( ( Xs = nil_b )
       => ( ( butlast_b @ ( cons_b @ X @ Xs ) )
          = nil_b ) )
      & ( ( Xs != nil_b )
       => ( ( butlast_b @ ( cons_b @ X @ Xs ) )
          = ( cons_b @ X @ ( butlast_b @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1674_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = nil_nat ) )
      & ( ( Xs != nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( butlast_nat @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1675_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_int,X: int] :
      ( ( ( Xs = nil_int )
       => ( ( butlast_int @ ( cons_int @ X @ Xs ) )
          = nil_int ) )
      & ( ( Xs != nil_int )
       => ( ( butlast_int @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( butlast_int @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1676_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( ( Xs = nil_Pr5671120429643327159n_assn )
       => ( ( butlas3012047794866324995n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = nil_Pr5671120429643327159n_assn ) )
      & ( ( Xs != nil_Pr5671120429643327159n_assn )
       => ( ( butlas3012047794866324995n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ X @ ( butlas3012047794866324995n_assn @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1677_butlast__append,axiom,
    ! [Ys: list_a,Xs: list_a] :
      ( ( ( Ys = nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
          = ( butlast_a @ Xs ) ) )
      & ( ( Ys != nil_a )
       => ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
          = ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1678_butlast__append,axiom,
    ! [Ys: list_b,Xs: list_b] :
      ( ( ( Ys = nil_b )
       => ( ( butlast_b @ ( append_b @ Xs @ Ys ) )
          = ( butlast_b @ Xs ) ) )
      & ( ( Ys != nil_b )
       => ( ( butlast_b @ ( append_b @ Xs @ Ys ) )
          = ( append_b @ Xs @ ( butlast_b @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1679_butlast__append,axiom,
    ! [Ys: list_P8527749157015355191n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( Ys = nil_Pr5671120429643327159n_assn )
       => ( ( butlas3012047794866324995n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
          = ( butlas3012047794866324995n_assn @ Xs ) ) )
      & ( ( Ys != nil_Pr5671120429643327159n_assn )
       => ( ( butlas3012047794866324995n_assn @ ( append282499809098378956n_assn @ Xs @ Ys ) )
          = ( append282499809098378956n_assn @ Xs @ ( butlas3012047794866324995n_assn @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1680_butlast__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( butlast_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs @ Ys ) )
          = ( append_nat @ Xs @ ( butlast_nat @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1681_butlast__append,axiom,
    ! [Ys: list_int,Xs: list_int] :
      ( ( ( Ys = nil_int )
       => ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
          = ( butlast_int @ Xs ) ) )
      & ( ( Ys != nil_int )
       => ( ( butlast_int @ ( append_int @ Xs @ Ys ) )
          = ( append_int @ Xs @ ( butlast_int @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1682_Inf__fin_Oin__idem,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A3 ) )
          = ( lattic956194824204696298_nat_o @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1683_Inf__fin_Oin__idem,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( member_assn @ X @ A3 )
       => ( ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) )
          = ( lattic47131356835913163n_assn @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1684_Inf__fin_Oin__idem,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
          = ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1685_Inf__fin_Oin__idem,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X @ A3 )
       => ( ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) )
          = ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1686_Inf__fin_Oin__idem,axiom,
    ! [A3: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( member_Product_unit @ X @ A3 )
       => ( ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) )
          = ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1687_Inf__fin_Oin__idem,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ X @ ( lattic30941717366863870at_nat @ A3 ) )
          = ( lattic30941717366863870at_nat @ A3 ) ) ) ) ).

% Inf_fin.in_idem
thf(fact_1688_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_a] :
      ( ( foldr_a_list_a @ cons_a @ Xs @ nil_a )
      = Xs ) ).

% Misc.foldr_Cons
thf(fact_1689_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_b] :
      ( ( foldr_b_list_b @ cons_b @ Xs @ nil_b )
      = Xs ) ).

% Misc.foldr_Cons
thf(fact_1690_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_nat] :
      ( ( foldr_nat_list_nat @ cons_nat @ Xs @ nil_nat )
      = Xs ) ).

% Misc.foldr_Cons
thf(fact_1691_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_int] :
      ( ( foldr_int_list_int @ cons_int @ Xs @ nil_int )
      = Xs ) ).

% Misc.foldr_Cons
thf(fact_1692_Misc_Ofoldr__Cons,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( foldr_2213011659765427309n_assn @ cons_P2971678138204555879n_assn @ Xs @ nil_Pr5671120429643327159n_assn )
      = Xs ) ).

% Misc.foldr_Cons
thf(fact_1693_Min__less__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ X )
          = ( ? [X3: $o] :
                ( ( member_o @ X3 @ A3 )
                & ( ord_less_o @ X3 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_1694_Min__less__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X )
          = ( ? [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
                & ( ord_less_nat @ X3 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_1695_Min__less__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ ( lattic8718645017227715691in_int @ A3 ) @ X )
          = ( ? [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
                & ( ord_less_int @ X3 @ X ) ) ) ) ) ) ).

% Min_less_iff
thf(fact_1696_Max__gr__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_o @ X @ ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ? [X3: $o] :
                ( ( member_o @ X3 @ A3 )
                & ( ord_less_o @ X @ X3 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_1697_Max__gr__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ? [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
                & ( ord_less_nat @ X @ X3 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_1698_Max__gr__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_int @ X @ ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ? [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
                & ( ord_less_int @ X @ X3 ) ) ) ) ) ) ).

% Max_gr_iff
thf(fact_1699_butlast__eq__consE,axiom,
    ! [L: list_a,X: a,Xs: list_a] :
      ( ( ( butlast_a @ L )
        = ( cons_a @ X @ Xs ) )
     => ~ ! [Xl: a] :
            ( L
           != ( cons_a @ X @ ( append_a @ Xs @ ( cons_a @ Xl @ nil_a ) ) ) ) ) ).

% butlast_eq_consE
thf(fact_1700_butlast__eq__consE,axiom,
    ! [L: list_b,X: b,Xs: list_b] :
      ( ( ( butlast_b @ L )
        = ( cons_b @ X @ Xs ) )
     => ~ ! [Xl: b] :
            ( L
           != ( cons_b @ X @ ( append_b @ Xs @ ( cons_b @ Xl @ nil_b ) ) ) ) ) ).

% butlast_eq_consE
thf(fact_1701_butlast__eq__consE,axiom,
    ! [L: list_nat,X: nat,Xs: list_nat] :
      ( ( ( butlast_nat @ L )
        = ( cons_nat @ X @ Xs ) )
     => ~ ! [Xl: nat] :
            ( L
           != ( cons_nat @ X @ ( append_nat @ Xs @ ( cons_nat @ Xl @ nil_nat ) ) ) ) ) ).

% butlast_eq_consE
thf(fact_1702_butlast__eq__consE,axiom,
    ! [L: list_int,X: int,Xs: list_int] :
      ( ( ( butlast_int @ L )
        = ( cons_int @ X @ Xs ) )
     => ~ ! [Xl: int] :
            ( L
           != ( cons_int @ X @ ( append_int @ Xs @ ( cons_int @ Xl @ nil_int ) ) ) ) ) ).

% butlast_eq_consE
thf(fact_1703_butlast__eq__consE,axiom,
    ! [L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( butlas3012047794866324995n_assn @ L )
        = ( cons_P2971678138204555879n_assn @ X @ Xs ) )
     => ~ ! [Xl: produc6575502325842934193n_assn] :
            ( L
           != ( cons_P2971678138204555879n_assn @ X @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Xl @ nil_Pr5671120429643327159n_assn ) ) ) ) ) ).

% butlast_eq_consE
thf(fact_1704_butlast__eq__cons__conv,axiom,
    ! [L: list_a,X: a,Xs: list_a] :
      ( ( ( butlast_a @ L )
        = ( cons_a @ X @ Xs ) )
      = ( ? [Xl2: a] :
            ( L
            = ( cons_a @ X @ ( append_a @ Xs @ ( cons_a @ Xl2 @ nil_a ) ) ) ) ) ) ).

% butlast_eq_cons_conv
thf(fact_1705_butlast__eq__cons__conv,axiom,
    ! [L: list_b,X: b,Xs: list_b] :
      ( ( ( butlast_b @ L )
        = ( cons_b @ X @ Xs ) )
      = ( ? [Xl2: b] :
            ( L
            = ( cons_b @ X @ ( append_b @ Xs @ ( cons_b @ Xl2 @ nil_b ) ) ) ) ) ) ).

% butlast_eq_cons_conv
thf(fact_1706_butlast__eq__cons__conv,axiom,
    ! [L: list_nat,X: nat,Xs: list_nat] :
      ( ( ( butlast_nat @ L )
        = ( cons_nat @ X @ Xs ) )
      = ( ? [Xl2: nat] :
            ( L
            = ( cons_nat @ X @ ( append_nat @ Xs @ ( cons_nat @ Xl2 @ nil_nat ) ) ) ) ) ) ).

% butlast_eq_cons_conv
thf(fact_1707_butlast__eq__cons__conv,axiom,
    ! [L: list_int,X: int,Xs: list_int] :
      ( ( ( butlast_int @ L )
        = ( cons_int @ X @ Xs ) )
      = ( ? [Xl2: int] :
            ( L
            = ( cons_int @ X @ ( append_int @ Xs @ ( cons_int @ Xl2 @ nil_int ) ) ) ) ) ) ).

% butlast_eq_cons_conv
thf(fact_1708_butlast__eq__cons__conv,axiom,
    ! [L: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( butlas3012047794866324995n_assn @ L )
        = ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( ? [Xl2: produc6575502325842934193n_assn] :
            ( L
            = ( cons_P2971678138204555879n_assn @ X @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ Xl2 @ nil_Pr5671120429643327159n_assn ) ) ) ) ) ) ).

% butlast_eq_cons_conv
thf(fact_1709_SuccD,axiom,
    ! [K: produc3658429121746597890et_nat > $o,Kl2: set_li630567559872716595_nat_o,Kl: list_P7985473006766602707_nat_o] :
      ( ( member6576561426505652726_nat_o @ K @ ( bNF_Gr992270151760408296_nat_o @ Kl2 @ Kl ) )
     => ( member4792007445727162748_nat_o @ ( append6126940767462283112_nat_o @ Kl @ ( cons_P6219271836124797827_nat_o @ K @ nil_Pr1626566771615838163_nat_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1710_SuccD,axiom,
    ! [K: a,Kl2: set_list_a,Kl: list_a] :
      ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
     => ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1711_SuccD,axiom,
    ! [K: b,Kl2: set_list_b,Kl: list_b] :
      ( ( member_b2 @ K @ ( bNF_Greatest_Succ_b @ Kl2 @ Kl ) )
     => ( member_list_b @ ( append_b @ Kl @ ( cons_b @ K @ nil_b ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1712_SuccD,axiom,
    ! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
      ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
     => ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1713_SuccD,axiom,
    ! [K: int,Kl2: set_list_int,Kl: list_int] :
      ( ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ Kl ) )
     => ( member_list_int @ ( append_int @ Kl @ ( cons_int @ K @ nil_int ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1714_SuccD,axiom,
    ! [K: produc6575502325842934193n_assn,Kl2: set_li5131720305576846103n_assn,Kl: list_P8527749157015355191n_assn] :
      ( ( member7957490590177025114n_assn @ K @ ( bNF_Gr3216292445876487756n_assn @ Kl2 @ Kl ) )
     => ( member852475432509897056n_assn @ ( append282499809098378956n_assn @ Kl @ ( cons_P2971678138204555879n_assn @ K @ nil_Pr5671120429643327159n_assn ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1715_empty__Shift,axiom,
    ! [Kl2: set_li630567559872716595_nat_o,K: produc3658429121746597890et_nat > $o] :
      ( ( member4792007445727162748_nat_o @ nil_Pr1626566771615838163_nat_o @ Kl2 )
     => ( ( member6576561426505652726_nat_o @ K @ ( bNF_Gr992270151760408296_nat_o @ Kl2 @ nil_Pr1626566771615838163_nat_o ) )
       => ( member4792007445727162748_nat_o @ nil_Pr1626566771615838163_nat_o @ ( bNF_Gr8793815481675871596_nat_o @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1716_empty__Shift,axiom,
    ! [Kl2: set_list_a,K: a] :
      ( ( member_list_a @ nil_a @ Kl2 )
     => ( ( member_a2 @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1717_empty__Shift,axiom,
    ! [Kl2: set_list_b,K: b] :
      ( ( member_list_b @ nil_b @ Kl2 )
     => ( ( member_b2 @ K @ ( bNF_Greatest_Succ_b @ Kl2 @ nil_b ) )
       => ( member_list_b @ nil_b @ ( bNF_Greatest_Shift_b @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1718_empty__Shift,axiom,
    ! [Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn] :
      ( ( member852475432509897056n_assn @ nil_Pr5671120429643327159n_assn @ Kl2 )
     => ( ( member7957490590177025114n_assn @ K @ ( bNF_Gr3216292445876487756n_assn @ Kl2 @ nil_Pr5671120429643327159n_assn ) )
       => ( member852475432509897056n_assn @ nil_Pr5671120429643327159n_assn @ ( bNF_Gr4113829767105464016n_assn @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1719_empty__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat] :
      ( ( member_list_nat @ nil_nat @ Kl2 )
     => ( ( member_nat2 @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1720_empty__Shift,axiom,
    ! [Kl2: set_list_int,K: int] :
      ( ( member_list_int @ nil_int @ Kl2 )
     => ( ( member_int2 @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl2 @ nil_int ) )
       => ( member_list_int @ nil_int @ ( bNF_Gr1870224194279859149ft_int @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1721_Succ__Shift,axiom,
    ! [Kl2: set_list_nat,K: nat,Kl: list_nat] :
      ( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ ( cons_nat @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_1722_Succ__Shift,axiom,
    ! [Kl2: set_list_int,K: int,Kl: list_int] :
      ( ( bNF_Gr6350390219475566417cc_int @ ( bNF_Gr1870224194279859149ft_int @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr6350390219475566417cc_int @ Kl2 @ ( cons_int @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_1723_Succ__Shift,axiom,
    ! [Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn,Kl: list_P8527749157015355191n_assn] :
      ( ( bNF_Gr3216292445876487756n_assn @ ( bNF_Gr4113829767105464016n_assn @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr3216292445876487756n_assn @ Kl2 @ ( cons_P2971678138204555879n_assn @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_1724_Sup__fin_Oinsert,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A3 ) )
          = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1725_Sup__fin_Oinsert,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A3 ) )
          = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1726_Sup__fin_Oinsert,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A3 ) )
          = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1727_Sup__fin_Oinsert,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A3 ) )
          = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).

% Sup_fin.insert
thf(fact_1728_Inf__fin_Oinsert,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A3 ) )
          = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1729_Inf__fin_Oinsert,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
          = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1730_Inf__fin_Oinsert,axiom,
    ! [A3: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( A3 != bot_bo3957492148770167129t_unit )
       => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
          = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1731_Inf__fin_Oinsert,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( A3 != bot_bo3083307316010499117at_nat )
       => ( ( lattic30941717366863870at_nat @ ( insert9200635055090092081at_nat @ X @ A3 ) )
          = ( inf_in2572325071724192079at_nat @ X @ ( lattic30941717366863870at_nat @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1732_Inf__fin_Oinsert,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A3 ) )
          = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1733_Inf__fin_Oinsert,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A3 ) )
          = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).

% Inf_fin.insert
thf(fact_1734_Inf__fin__le__Sup__fin,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ A3 ) @ ( lattic1508158080041050831_fin_o @ A3 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1735_Inf__fin__le__Sup__fin,axiom,
    ! [A3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ord_less_eq_assn @ ( lattic47131356835913163n_assn @ A3 ) @ ( lattic2150320897289308081n_assn @ A3 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1736_Inf__fin__le__Sup__fin,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1737_Inf__fin__le__Sup__fin,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ A3 ) @ ( lattic1091506334969745077in_int @ A3 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1738_Inf__fin__le__Sup__fin,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ A3 ) @ ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ).

% Inf_fin_le_Sup_fin
thf(fact_1739_Sup__fin_Osubset,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( B3 != bot_bot_set_assn )
       => ( ( ord_less_eq_set_assn @ B3 @ A3 )
         => ( ( sup_sup_assn @ ( lattic2150320897289308081n_assn @ B3 ) @ ( lattic2150320897289308081n_assn @ A3 ) )
            = ( lattic2150320897289308081n_assn @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1740_Sup__fin_Osubset,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( B3 != bot_bot_set_set_nat )
       => ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
         => ( ( sup_sup_set_nat @ ( lattic3835124923745554447et_nat @ B3 ) @ ( lattic3835124923745554447et_nat @ A3 ) )
            = ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1741_Sup__fin_Osubset,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( B3 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B3 @ A3 )
         => ( ( sup_sup_o @ ( lattic1508158080041050831_fin_o @ B3 ) @ ( lattic1508158080041050831_fin_o @ A3 ) )
            = ( lattic1508158080041050831_fin_o @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1742_Sup__fin_Osubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A3 )
         => ( ( sup_sup_nat @ ( lattic1093996805478795353in_nat @ B3 ) @ ( lattic1093996805478795353in_nat @ A3 ) )
            = ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ).

% Sup_fin.subset
thf(fact_1743_Inf__fin_Osubset,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( B3 != bot_bot_set_assn )
       => ( ( ord_less_eq_set_assn @ B3 @ A3 )
         => ( ( inf_inf_assn @ ( lattic47131356835913163n_assn @ B3 ) @ ( lattic47131356835913163n_assn @ A3 ) )
            = ( lattic47131356835913163n_assn @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1744_Inf__fin_Osubset,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( B3 != bot_bot_set_set_nat )
       => ( ( ord_le6893508408891458716et_nat @ B3 @ A3 )
         => ( ( inf_inf_set_nat @ ( lattic3014633134055518761et_nat @ B3 ) @ ( lattic3014633134055518761et_nat @ A3 ) )
            = ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1745_Inf__fin_Osubset,axiom,
    ! [A3: set_Product_unit,B3: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( B3 != bot_bo3957492148770167129t_unit )
       => ( ( ord_le3507040750410214029t_unit @ B3 @ A3 )
         => ( ( inf_inf_Product_unit @ ( lattic1263872656861969706t_unit @ B3 ) @ ( lattic1263872656861969706t_unit @ A3 ) )
            = ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1746_Inf__fin_Osubset,axiom,
    ! [A3: set_se7855581050983116737at_nat,B3: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( B3 != bot_bo3083307316010499117at_nat )
       => ( ( ord_le2077887516847798113at_nat @ B3 @ A3 )
         => ( ( inf_in2572325071724192079at_nat @ ( lattic30941717366863870at_nat @ B3 ) @ ( lattic30941717366863870at_nat @ A3 ) )
            = ( lattic30941717366863870at_nat @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1747_Inf__fin_Osubset,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( B3 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B3 @ A3 )
         => ( ( inf_inf_o @ ( lattic4107685809792843317_fin_o @ B3 ) @ ( lattic4107685809792843317_fin_o @ A3 ) )
            = ( lattic4107685809792843317_fin_o @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1748_Inf__fin_Osubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A3 )
         => ( ( inf_inf_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A3 ) )
            = ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).

% Inf_fin.subset
thf(fact_1749_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( A3 != bot_bo7824918357723582233_nat_o )
         => ( ( lattic7320199455484906628_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) )
            = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1750_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ~ ( member_assn @ X @ A3 )
       => ( ( A3 != bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A3 ) )
            = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1751_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ~ ( member_set_nat @ X @ A3 )
       => ( ( A3 != bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A3 ) )
            = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1752_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A3 ) )
            = ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1753_Sup__fin_Oinsert__not__elem,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat2 @ X @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A3 ) ) ) ) ) ) ).

% Sup_fin.insert_not_elem
thf(fact_1754_Sup__fin_Oclosed,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( sup_su5453871518329203617_nat_o @ X2 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X2 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
         => ( member6576561426505652726_nat_o @ ( lattic7320199455484906628_nat_o @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1755_Sup__fin_Oclosed,axiom,
    ! [A3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ! [X2: assn,Y2: assn] : ( member_assn @ ( sup_sup_assn @ X2 @ Y2 ) @ ( insert_assn @ X2 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
         => ( member_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1756_Sup__fin_Oclosed,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y2: set_nat] : ( member_set_nat @ ( sup_sup_set_nat @ X2 @ Y2 ) @ ( insert_set_nat @ X2 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
         => ( member_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1757_Sup__fin_Oclosed,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : ( member_o @ ( sup_sup_o @ X2 @ Y2 ) @ ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1758_Sup__fin_Oclosed,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( sup_sup_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
         => ( member_nat2 @ ( lattic1093996805478795353in_nat @ A3 ) @ A3 ) ) ) ) ).

% Sup_fin.closed
thf(fact_1759_order__refl,axiom,
    ! [X: assn] : ( ord_less_eq_assn @ X @ X ) ).

% order_refl
thf(fact_1760_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_1761_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_1762_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_1763_dual__order_Orefl,axiom,
    ! [A: assn] : ( ord_less_eq_assn @ A @ A ) ).

% dual_order.refl
thf(fact_1764_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1765_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_1766_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_1767_subsetI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X2 @ A3 )
         => ( member6576561426505652726_nat_o @ X2 @ B3 ) )
     => ( ord_le2965882846123202637_nat_o @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1768_subsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A3 )
         => ( member_nat2 @ X2 @ B3 ) )
     => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1769_subset__antisym,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% subset_antisym
thf(fact_1770_insertCI,axiom,
    ! [A: $o,B3: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B3 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B3 ) ) ) ).

% insertCI
thf(fact_1771_insertCI,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
      ( ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
       => ( A = B ) )
     => ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ B3 ) ) ) ).

% insertCI
thf(fact_1772_insert__iff,axiom,
    ! [A: $o,B: $o,A3: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A3 ) )
      = ( ( A = B )
        | ( member_o @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1773_insert__iff,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ A3 ) )
      = ( ( A = B )
        | ( member6576561426505652726_nat_o @ A @ A3 ) ) ) ).

% insert_iff
thf(fact_1774_insert__absorb2,axiom,
    ! [X: $o,A3: set_o] :
      ( ( insert_o @ X @ ( insert_o @ X @ A3 ) )
      = ( insert_o @ X @ A3 ) ) ).

% insert_absorb2
thf(fact_1775_inf_Obounded__iff,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) )
      = ( ( ord_le3221252021190050221t_unit @ A @ B )
        & ( ord_le3221252021190050221t_unit @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1776_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1777_inf_Obounded__iff,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
      = ( ( ord_less_eq_assn @ A @ B )
        & ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1778_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1779_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1780_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_1781_le__inf__iff,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) )
      = ( ( ord_le3221252021190050221t_unit @ X @ Y )
        & ( ord_le3221252021190050221t_unit @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1782_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) )
      = ( ( ord_le3146513528884898305at_nat @ X @ Y )
        & ( ord_le3146513528884898305at_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1783_le__inf__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) )
      = ( ( ord_less_eq_assn @ X @ Y )
        & ( ord_less_eq_assn @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1784_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1785_le__inf__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1786_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% le_inf_iff
thf(fact_1787_le__sup__iff,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ X @ Y ) @ Z )
      = ( ( ord_less_eq_assn @ X @ Z )
        & ( ord_less_eq_assn @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1788_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        & ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1789_le__sup__iff,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        & ( ord_less_eq_int @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1790_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_set_nat @ X @ Z )
        & ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).

% le_sup_iff
thf(fact_1791_sup_Obounded__iff,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C2 ) @ A )
      = ( ( ord_less_eq_assn @ B @ A )
        & ( ord_less_eq_assn @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1792_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1793_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1794_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1795_empty__subsetI,axiom,
    ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).

% empty_subsetI
thf(fact_1796_empty__subsetI,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).

% empty_subsetI
thf(fact_1797_subset__empty,axiom,
    ! [A3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
      = ( A3 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_1798_subset__empty,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
      = ( A3 = bot_bot_set_nat ) ) ).

% subset_empty
thf(fact_1799_singletonI,axiom,
    ! [A: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) ).

% singletonI
thf(fact_1800_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_1801_singletonI,axiom,
    ! [A: nat] : ( member_nat2 @ A @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_1802_insert__subset,axiom,
    ! [X: $o,A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X @ A3 ) @ B3 )
      = ( ( member_o @ X @ B3 )
        & ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ).

% insert_subset
thf(fact_1803_insert__subset,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) @ B3 )
      = ( ( member6576561426505652726_nat_o @ X @ B3 )
        & ( ord_le2965882846123202637_nat_o @ A3 @ B3 ) ) ) ).

% insert_subset
thf(fact_1804_insert__subset,axiom,
    ! [X: nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat2 @ X @ A3 ) @ B3 )
      = ( ( member_nat2 @ X @ B3 )
        & ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).

% insert_subset
thf(fact_1805_Int__subset__iff,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = ( ( ord_le3146513528884898305at_nat @ C3 @ A3 )
        & ( ord_le3146513528884898305at_nat @ C3 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1806_Int__subset__iff,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = ( ( ord_less_eq_set_nat @ C3 @ A3 )
        & ( ord_less_eq_set_nat @ C3 @ B3 ) ) ) ).

% Int_subset_iff
thf(fact_1807_Int__insert__right__if1,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1808_Int__insert__right__if1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
        = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1809_Int__insert__right__if1,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
        = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1810_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_1811_Int__insert__right__if0,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ~ ( member_o @ A @ A3 )
     => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
        = ( inf_inf_set_o @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1812_Int__insert__right__if0,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
        = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1813_Int__insert__right__if0,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ~ ( member_nat2 @ A @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
        = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1814_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
        = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ).

% Int_insert_right_if0
thf(fact_1815_insert__inter__insert,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ ( insert_o @ A @ B3 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_1816_insert__inter__insert,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ ( insert_nat2 @ A @ B3 ) )
      = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_1817_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ ( insert8211810215607154385at_nat @ A @ B3 ) )
      = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ).

% insert_inter_insert
thf(fact_1818_Int__insert__left__if1,axiom,
    ! [A: $o,C3: set_o,B3: set_o] :
      ( ( member_o @ A @ C3 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C3 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B3 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1819_Int__insert__left__if1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ C3 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C3 )
        = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B3 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1820_Int__insert__left__if1,axiom,
    ! [A: nat,C3: set_nat,B3: set_nat] :
      ( ( member_nat2 @ A @ C3 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C3 )
        = ( insert_nat2 @ A @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1821_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ C3 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C3 )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_1822_Int__insert__left__if0,axiom,
    ! [A: $o,C3: set_o,B3: set_o] :
      ( ~ ( member_o @ A @ C3 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C3 )
        = ( inf_inf_set_o @ B3 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1823_Int__insert__left__if0,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A @ C3 )
     => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C3 )
        = ( inf_in1906310914598751387_nat_o @ B3 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1824_Int__insert__left__if0,axiom,
    ! [A: nat,C3: set_nat,B3: set_nat] :
      ( ~ ( member_nat2 @ A @ C3 )
     => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C3 )
        = ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1825_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ C3 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C3 )
        = ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ).

% Int_insert_left_if0
thf(fact_1826_Un__subset__iff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( ( ord_less_eq_set_nat @ A3 @ C3 )
        & ( ord_less_eq_set_nat @ B3 @ C3 ) ) ) ).

% Un_subset_iff
thf(fact_1827_Un__insert__left,axiom,
    ! [A: $o,B3: set_o,C3: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B3 ) @ C3 )
      = ( insert_o @ A @ ( sup_sup_set_o @ B3 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_1828_Un__insert__left,axiom,
    ! [A: nat,B3: set_nat,C3: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat2 @ A @ B3 ) @ C3 )
      = ( insert_nat2 @ A @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% Un_insert_left
thf(fact_1829_Un__insert__right,axiom,
    ! [A3: set_o,A: $o,B3: set_o] :
      ( ( sup_sup_set_o @ A3 @ ( insert_o @ A @ B3 ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A3 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_1830_Un__insert__right,axiom,
    ! [A3: set_nat,A: nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
      = ( insert_nat2 @ A @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% Un_insert_right
thf(fact_1831_psubsetI,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != B3 )
       => ( ord_less_set_nat @ A3 @ B3 ) ) ) ).

% psubsetI
thf(fact_1832_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A3: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1833_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A3: set_nat] :
      ( ( ( insert_nat2 @ B @ bot_bot_set_nat )
        = ( insert_nat2 @ A @ A3 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_1834_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A3: set_o,B: $o] :
      ( ( ( insert_o @ A @ A3 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1835_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A3: set_nat,B: nat] :
      ( ( ( insert_nat2 @ A @ A3 )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_1836_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ B @ B3 ) ) )
      = ( ~ ( member6576561426505652726_nat_o @ B @ A3 )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1837_disjoint__insert_I2_J,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ B @ B3 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A3 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1838_disjoint__insert_I2_J,axiom,
    ! [A3: set_o,B: $o,B3: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A3 @ ( insert_o @ B @ B3 ) ) )
      = ( ~ ( member_o @ B @ A3 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1839_disjoint__insert_I2_J,axiom,
    ! [A3: set_nat,B: nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ B @ B3 ) ) )
      = ( ~ ( member_nat2 @ B @ A3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_1840_disjoint__insert_I1_J,axiom,
    ! [B3: set_Pr4532377907799695533_nat_o,A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ B3 @ ( insert5175938949040314269_nat_o @ A @ A3 ) )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( ( inf_in1906310914598751387_nat_o @ B3 @ A3 )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1841_disjoint__insert_I1_J,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B3 @ ( insert8211810215607154385at_nat @ A @ A3 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( ( inf_in2572325071724192079at_nat @ B3 @ A3 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1842_disjoint__insert_I1_J,axiom,
    ! [B3: set_o,A: $o,A3: set_o] :
      ( ( ( inf_inf_set_o @ B3 @ ( insert_o @ A @ A3 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B3 )
        & ( ( inf_inf_set_o @ B3 @ A3 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_1843_disjoint__insert_I1_J,axiom,
    ! [B3: set_nat,A: nat,A3: set_nat] :
      ( ( ( inf_inf_set_nat @ B3 @ ( insert_nat2 @ A @ A3 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( ( inf_inf_set_nat @ B3 @ A3 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_1844_insert__disjoint_I2_J,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( bot_bo7824918357723582233_nat_o
        = ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A3 ) @ B3 ) )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( bot_bo7824918357723582233_nat_o
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1845_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ B3 ) )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1846_insert__disjoint_I2_J,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B3 ) )
      = ( ~ ( member_o @ A @ B3 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1847_insert__disjoint_I2_J,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B3 ) )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_1848_insert__disjoint_I1_J,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ A3 ) @ B3 )
        = bot_bo7824918357723582233_nat_o )
      = ( ~ ( member6576561426505652726_nat_o @ A @ B3 )
        & ( ( inf_in1906310914598751387_nat_o @ A3 @ B3 )
          = bot_bo7824918357723582233_nat_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1849_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A3 ) @ B3 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B3 )
        & ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1850_insert__disjoint_I1_J,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A3 ) @ B3 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B3 )
        & ( ( inf_inf_set_o @ A3 @ B3 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_1851_insert__disjoint_I1_J,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ A3 ) @ B3 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat2 @ A @ B3 )
        & ( ( inf_inf_set_nat @ A3 @ B3 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_1852_Max__singleton,axiom,
    ! [X: $o] :
      ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Max_singleton
thf(fact_1853_Max__singleton,axiom,
    ! [X: nat] :
      ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% Max_singleton
thf(fact_1854_Min__singleton,axiom,
    ! [X: $o] :
      ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Min_singleton
thf(fact_1855_Min__singleton,axiom,
    ! [X: nat] :
      ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% Min_singleton
thf(fact_1856_Sup__fin_Osingleton,axiom,
    ! [X: $o] :
      ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_1857_Sup__fin_Osingleton,axiom,
    ! [X: nat] :
      ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% Sup_fin.singleton
thf(fact_1858_Inf__fin_Osingleton,axiom,
    ! [X: $o] :
      ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% Inf_fin.singleton
thf(fact_1859_Inf__fin_Osingleton,axiom,
    ! [X: nat] :
      ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% Inf_fin.singleton
thf(fact_1860_Max_Obounded__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X3 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1861_Max_Obounded__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1862_Max_Obounded__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X3 @ X ) ) ) ) ) ) ).

% Max.bounded_iff
thf(fact_1863_Min_Obounded__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X @ X3 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1864_Min_Obounded__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1865_Min_Obounded__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A3 ) )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X @ X3 ) ) ) ) ) ) ).

% Min.bounded_iff
thf(fact_1866_in__mono,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ B3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( member6576561426505652726_nat_o @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1867_in__mono,axiom,
    ! [A3: set_nat,B3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( member_nat2 @ X @ B3 ) ) ) ).

% in_mono
thf(fact_1868_insertE,axiom,
    ! [A: $o,B: $o,A3: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A3 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A3 ) ) ) ).

% insertE
thf(fact_1869_insertE,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ A3 ) )
     => ( ( A != B )
       => ( member6576561426505652726_nat_o @ A @ A3 ) ) ) ).

% insertE
thf(fact_1870_subsetD,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,C2: produc3658429121746597890et_nat > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ B3 )
     => ( ( member6576561426505652726_nat_o @ C2 @ A3 )
       => ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_1871_subsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( member_nat2 @ C2 @ A3 )
       => ( member_nat2 @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_1872_insertI1,axiom,
    ! [A: $o,B3: set_o] : ( member_o @ A @ ( insert_o @ A @ B3 ) ) ).

% insertI1
thf(fact_1873_insertI1,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] : ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ A @ B3 ) ) ).

% insertI1
thf(fact_1874_insertI2,axiom,
    ! [A: $o,B3: set_o,B: $o] :
      ( ( member_o @ A @ B3 )
     => ( member_o @ A @ ( insert_o @ B @ B3 ) ) ) ).

% insertI2
thf(fact_1875_insertI2,axiom,
    ! [A: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ A @ B3 )
     => ( member6576561426505652726_nat_o @ A @ ( insert5175938949040314269_nat_o @ B @ B3 ) ) ) ).

% insertI2
thf(fact_1876_equalityE,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( A3 = B3 )
     => ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ~ ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).

% equalityE
thf(fact_1877_subset__eq,axiom,
    ( ord_le2965882846123202637_nat_o
    = ( ^ [A5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
        ! [X3: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ X3 @ A5 )
         => ( member6576561426505652726_nat_o @ X3 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_1878_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
        ! [X3: nat] :
          ( ( member_nat2 @ X3 @ A5 )
         => ( member_nat2 @ X3 @ B4 ) ) ) ) ).

% subset_eq
thf(fact_1879_equalityD1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( A3 = B3 )
     => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% equalityD1
thf(fact_1880_equalityD2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( A3 = B3 )
     => ( ord_less_eq_set_nat @ B3 @ A3 ) ) ).

% equalityD2
thf(fact_1881_Set_Oset__insert,axiom,
    ! [X: $o,A3: set_o] :
      ( ( member_o @ X @ A3 )
     => ~ ! [B6: set_o] :
            ( ( A3
              = ( insert_o @ X @ B6 ) )
           => ( member_o @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1882_Set_Oset__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ A3 )
     => ~ ! [B6: set_Pr4532377907799695533_nat_o] :
            ( ( A3
              = ( insert5175938949040314269_nat_o @ X @ B6 ) )
           => ( member6576561426505652726_nat_o @ X @ B6 ) ) ) ).

% Set.set_insert
thf(fact_1883_subset__iff,axiom,
    ( ord_le2965882846123202637_nat_o
    = ( ^ [A5: set_Pr4532377907799695533_nat_o,B4: set_Pr4532377907799695533_nat_o] :
        ! [T2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ T2 @ A5 )
         => ( member6576561426505652726_nat_o @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_1884_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
        ! [T2: nat] :
          ( ( member_nat2 @ T2 @ A5 )
         => ( member_nat2 @ T2 @ B4 ) ) ) ) ).

% subset_iff
thf(fact_1885_insert__mono,axiom,
    ! [C3: set_o,D: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C3 @ D )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C3 ) @ ( insert_o @ A @ D ) ) ) ).

% insert_mono
thf(fact_1886_insert__mono,axiom,
    ! [C3: set_nat,D: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C3 @ D )
     => ( ord_less_eq_set_nat @ ( insert_nat2 @ A @ C3 ) @ ( insert_nat2 @ A @ D ) ) ) ).

% insert_mono
thf(fact_1887_subset__refl,axiom,
    ! [A3: set_nat] : ( ord_less_eq_set_nat @ A3 @ A3 ) ).

% subset_refl
thf(fact_1888_Collect__mono,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q ) ) ) ).

% Collect_mono
thf(fact_1889_Collect__mono,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1890_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X2: nat] :
          ( ( P @ X2 )
         => ( Q @ X2 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1891_insert__ident,axiom,
    ! [X: $o,A3: set_o,B3: set_o] :
      ( ~ ( member_o @ X @ A3 )
     => ( ~ ( member_o @ X @ B3 )
       => ( ( ( insert_o @ X @ A3 )
            = ( insert_o @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_1892_insert__ident,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ X @ B3 )
       => ( ( ( insert5175938949040314269_nat_o @ X @ A3 )
            = ( insert5175938949040314269_nat_o @ X @ B3 ) )
          = ( A3 = B3 ) ) ) ) ).

% insert_ident
thf(fact_1893_subset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ord_less_eq_set_nat @ A3 @ C3 ) ) ) ).

% subset_trans
thf(fact_1894_insert__absorb,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( insert_o @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_1895_insert__absorb,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( insert5175938949040314269_nat_o @ A @ A3 )
        = A3 ) ) ).

% insert_absorb
thf(fact_1896_insert__eq__iff,axiom,
    ! [A: $o,A3: set_o,B: $o,B3: set_o] :
      ( ~ ( member_o @ A @ A3 )
     => ( ~ ( member_o @ B @ B3 )
       => ( ( ( insert_o @ A @ A3 )
            = ( insert_o @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A = ~ B )
             => ? [C4: set_o] :
                  ( ( A3
                    = ( insert_o @ B @ C4 ) )
                  & ~ ( member_o @ B @ C4 )
                  & ( B3
                    = ( insert_o @ A @ C4 ) )
                  & ~ ( member_o @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1897_insert__eq__iff,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ B @ B3 )
       => ( ( ( insert5175938949040314269_nat_o @ A @ A3 )
            = ( insert5175938949040314269_nat_o @ B @ B3 ) )
          = ( ( ( A = B )
             => ( A3 = B3 ) )
            & ( ( A != B )
             => ? [C4: set_Pr4532377907799695533_nat_o] :
                  ( ( A3
                    = ( insert5175938949040314269_nat_o @ B @ C4 ) )
                  & ~ ( member6576561426505652726_nat_o @ B @ C4 )
                  & ( B3
                    = ( insert5175938949040314269_nat_o @ A @ C4 ) )
                  & ~ ( member6576561426505652726_nat_o @ A @ C4 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_1898_set__eq__subset,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : Y5 = Z4 )
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_1899_subset__insert,axiom,
    ! [X: $o,A3: set_o,B3: set_o] :
      ( ~ ( member_o @ X @ A3 )
     => ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X @ B3 ) )
        = ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ).

% subset_insert
thf(fact_1900_subset__insert,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
     => ( ( ord_le2965882846123202637_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ B3 ) )
        = ( ord_le2965882846123202637_nat_o @ A3 @ B3 ) ) ) ).

% subset_insert
thf(fact_1901_subset__insert,axiom,
    ! [X: nat,A3: set_nat,B3: set_nat] :
      ( ~ ( member_nat2 @ X @ A3 )
     => ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X @ B3 ) )
        = ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ).

% subset_insert
thf(fact_1902_insert__commute,axiom,
    ! [X: $o,Y: $o,A3: set_o] :
      ( ( insert_o @ X @ ( insert_o @ Y @ A3 ) )
      = ( insert_o @ Y @ ( insert_o @ X @ A3 ) ) ) ).

% insert_commute
thf(fact_1903_subset__insertI,axiom,
    ! [B3: set_o,A: $o] : ( ord_less_eq_set_o @ B3 @ ( insert_o @ A @ B3 ) ) ).

% subset_insertI
thf(fact_1904_subset__insertI,axiom,
    ! [B3: set_nat,A: nat] : ( ord_less_eq_set_nat @ B3 @ ( insert_nat2 @ A @ B3 ) ) ).

% subset_insertI
thf(fact_1905_subset__insertI2,axiom,
    ! [A3: set_o,B3: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_1906_subset__insertI2,axiom,
    ! [A3: set_nat,B3: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ B @ B3 ) ) ) ).

% subset_insertI2
thf(fact_1907_Collect__mono__iff,axiom,
    ! [P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ ( collec939566748876313656_nat_o @ P ) @ ( collec939566748876313656_nat_o @ Q ) )
      = ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1908_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1909_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q @ X3 ) ) ) ) ).

% Collect_mono_iff
thf(fact_1910_mk__disjoint__insert,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ? [B6: set_o] :
          ( ( A3
            = ( insert_o @ A @ B6 ) )
          & ~ ( member_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1911_mk__disjoint__insert,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ A3 )
     => ? [B6: set_Pr4532377907799695533_nat_o] :
          ( ( A3
            = ( insert5175938949040314269_nat_o @ A @ B6 ) )
          & ~ ( member6576561426505652726_nat_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_1912_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1913_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_1914_le__cases3,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z ) )
       => ( ( ( ord_less_eq_nat @ X @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1915_le__cases3,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z ) )
       => ( ( ( ord_less_eq_int @ X @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X ) )
             => ~ ( ( ord_less_eq_int @ Z @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1916_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: assn,Z4: assn] : Y5 = Z4 )
    = ( ^ [X3: assn,Y3: assn] :
          ( ( ord_less_eq_assn @ X3 @ Y3 )
          & ( ord_less_eq_assn @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1917_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1918_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1919_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : Y5 = Z4 )
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y3 )
          & ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1920_ord__eq__le__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( A = B )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1921_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1922_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1923_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_1924_ord__le__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1925_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1926_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1927_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_1928_order__antisym,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1929_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1930_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1931_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_1932_order_Otrans,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% order.trans
thf(fact_1933_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_1934_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_1935_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_1936_order__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_eq_assn @ X @ Z ) ) ) ).

% order_trans
thf(fact_1937_order__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1938_order__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X @ Z ) ) ) ).

% order_trans
thf(fact_1939_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_eq_set_nat @ X @ Z ) ) ) ).

% order_trans
thf(fact_1940_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A4 @ B5 )
         => ( P @ A4 @ B5 ) )
     => ( ! [A4: nat,B5: nat] :
            ( ( P @ B5 @ A4 )
           => ( P @ A4 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1941_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A4: int,B5: int] :
          ( ( ord_less_eq_int @ A4 @ B5 )
         => ( P @ A4 @ B5 ) )
     => ( ! [A4: int,B5: int] :
            ( ( P @ B5 @ A4 )
           => ( P @ A4 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1942_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: assn,Z4: assn] : Y5 = Z4 )
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ B2 @ A2 )
          & ( ord_less_eq_assn @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1943_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1944_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1945_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : Y5 = Z4 )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1946_dual__order_Oantisym,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_eq_assn @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1947_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1948_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1949_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_1950_dual__order_Otrans,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C2 @ B )
       => ( ord_less_eq_assn @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_1951_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_1952_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_1953_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ B )
       => ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_1954_antisym,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1955_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1956_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1957_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1958_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: assn,Z4: assn] : Y5 = Z4 )
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ( ord_less_eq_assn @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1959_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z4: nat] : Y5 = Z4 )
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1960_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1961_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z4: set_nat] : Y5 = Z4 )
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1962_order__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1963_order__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1964_order__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1965_order__subst1,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1966_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1967_order__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1968_order__subst1,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1969_order__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1970_order__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1971_order__subst1,axiom,
    ! [A: assn,F: set_nat > assn,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ! [X2: set_nat,Y2: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_1972_order__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1973_order__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1974_order__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1975_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1976_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1977_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1978_order__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1979_order__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1980_order__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1981_order__subst2,axiom,
    ! [A: assn,B: assn,F: assn > set_nat,C2: set_nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_1982_ord__eq__le__eq__trans,axiom,
    ! [A: assn,B: assn,C2: assn,D2: assn] :
      ( ( A = B )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_assn @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1983_ord__eq__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1984_ord__eq__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_int @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1985_ord__eq__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D2: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ( C2 = D2 )
         => ( ord_less_eq_set_nat @ A @ D2 ) ) ) ) ).

% ord_eq_le_eq_trans
thf(fact_1986_order__eq__refl,axiom,
    ! [X: assn,Y: assn] :
      ( ( X = Y )
     => ( ord_less_eq_assn @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1987_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1988_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1989_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_1990_subset__Collect__conv,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ S @ ( collec939566748876313656_nat_o @ P ) )
      = ( ! [X3: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1991_subset__Collect__conv,axiom,
    ! [S: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ S @ ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1992_subset__Collect__conv,axiom,
    ! [S: set_nat,P: nat > $o] :
      ( ( ord_less_eq_set_nat @ S @ ( collect_nat @ P ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ S )
           => ( P @ X3 ) ) ) ) ).

% subset_Collect_conv
thf(fact_1993_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_1994_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_1995_ord__eq__le__subst,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1996_ord__eq__le__subst,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1997_ord__eq__le__subst,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1998_ord__eq__le__subst,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1999_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2000_ord__eq__le__subst,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2001_ord__eq__le__subst,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2002_ord__eq__le__subst,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2003_ord__eq__le__subst,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2004_ord__eq__le__subst,axiom,
    ! [A: set_nat,F: assn > set_nat,B: assn,C2: assn] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_2005_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2006_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2007_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2008_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2009_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2010_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2011_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2012_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2013_ord__le__eq__subst,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2014_ord__le__eq__subst,axiom,
    ! [A: assn,B: assn,F: assn > set_nat,C2: set_nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_eq_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_2015_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_2016_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_2017_order__antisym__conv,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_2018_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_2019_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_2020_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_2021_subset__singletonD,axiom,
    ! [A3: set_o,X: $o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
     => ( ( A3 = bot_bot_set_o )
        | ( A3
          = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_2022_subset__singletonD,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
     => ( ( A3 = bot_bot_set_nat )
        | ( A3
          = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_2023_subset__singleton__iff,axiom,
    ! [X5: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X5 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X5 = bot_bot_set_o )
        | ( X5
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_2024_subset__singleton__iff,axiom,
    ! [X5: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X5 @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
      = ( ( X5 = bot_bot_set_nat )
        | ( X5
          = ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_2025_finite__ranking__induct,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o,F: ( produc3658429121746597890et_nat > $o ) > nat] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,S2: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ S2 )
             => ( ! [Y4: produc3658429121746597890et_nat > $o] :
                    ( ( member6576561426505652726_nat_o @ Y4 @ S2 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2026_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X2: $o,S2: set_o] :
              ( ( finite_finite_o @ S2 )
             => ( ! [Y4: $o] :
                    ( ( member_o @ Y4 @ S2 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_o @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2027_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,S2: set_nat] :
              ( ( finite_finite_nat @ S2 )
             => ( ! [Y4: nat] :
                    ( ( member_nat2 @ Y4 @ S2 )
                   => ( ord_less_eq_nat @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_nat2 @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2028_finite__ranking__induct,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o,F: ( produc3658429121746597890et_nat > $o ) > int] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,S2: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ S2 )
             => ( ! [Y4: produc3658429121746597890et_nat > $o] :
                    ( ( member6576561426505652726_nat_o @ Y4 @ S2 )
                   => ( ord_less_eq_int @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2029_finite__ranking__induct,axiom,
    ! [S: set_o,P: set_o > $o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X2: $o,S2: set_o] :
              ( ( finite_finite_o @ S2 )
             => ( ! [Y4: $o] :
                    ( ( member_o @ Y4 @ S2 )
                   => ( ord_less_eq_int @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_o @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2030_finite__ranking__induct,axiom,
    ! [S: set_nat,P: set_nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,S2: set_nat] :
              ( ( finite_finite_nat @ S2 )
             => ( ! [Y4: nat] :
                    ( ( member_nat2 @ Y4 @ S2 )
                   => ( ord_less_eq_int @ ( F @ Y4 ) @ ( F @ X2 ) ) )
               => ( ( P @ S2 )
                 => ( P @ ( insert_nat2 @ X2 @ S2 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_ranking_induct
thf(fact_2031_finite__subset__induct_H,axiom,
    ! [F2: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F2 )
     => ( ( ord_le2965882846123202637_nat_o @ F2 @ A3 )
       => ( ( P @ bot_bo7824918357723582233_nat_o )
         => ( ! [A4: produc3658429121746597890et_nat > $o,F4: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F4 )
               => ( ( member6576561426505652726_nat_o @ A4 @ A3 )
                 => ( ( ord_le2965882846123202637_nat_o @ F4 @ A3 )
                   => ( ~ ( member6576561426505652726_nat_o @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2032_finite__subset__induct_H,axiom,
    ! [F2: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A4: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A4 @ A3 )
                 => ( ( ord_less_eq_set_o @ F4 @ A3 )
                   => ( ~ ( member_o @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_o @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2033_finite__subset__induct_H,axiom,
    ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat2 @ A4 @ A3 )
                 => ( ( ord_less_eq_set_nat @ F4 @ A3 )
                   => ( ~ ( member_nat2 @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert_nat2 @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_2034_finite__subset__induct,axiom,
    ! [F2: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F2 )
     => ( ( ord_le2965882846123202637_nat_o @ F2 @ A3 )
       => ( ( P @ bot_bo7824918357723582233_nat_o )
         => ( ! [A4: produc3658429121746597890et_nat > $o,F4: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F4 )
               => ( ( member6576561426505652726_nat_o @ A4 @ A3 )
                 => ( ~ ( member6576561426505652726_nat_o @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert5175938949040314269_nat_o @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2035_finite__subset__induct,axiom,
    ! [F2: set_o,A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( ord_less_eq_set_o @ F2 @ A3 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A4: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( member_o @ A4 @ A3 )
                 => ( ~ ( member_o @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2036_finite__subset__induct,axiom,
    ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( ord_less_eq_set_nat @ F2 @ A3 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A4: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( member_nat2 @ A4 @ A3 )
                 => ( ~ ( member_nat2 @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat2 @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_2037_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2038_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat2 @ A @ bot_bot_set_nat )
        = ( insert_nat2 @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_2039_insert__not__empty,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ A3 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_2040_insert__not__empty,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat2 @ A @ A3 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_2041_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C2: $o,D2: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C2 @ ( insert_o @ D2 @ bot_bot_set_o ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2042_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( insert_nat2 @ A @ ( insert_nat2 @ B @ bot_bot_set_nat ) )
        = ( insert_nat2 @ C2 @ ( insert_nat2 @ D2 @ bot_bot_set_nat ) ) )
      = ( ( ( A = C2 )
          & ( B = D2 ) )
        | ( ( A = D2 )
          & ( B = C2 ) ) ) ) ).

% doubleton_eq_iff
thf(fact_2043_singleton__iff,axiom,
    ! [B: produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2044_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2045_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_2046_singletonD,axiom,
    ! [B: produc3658429121746597890et_nat > $o,A: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ B @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2047_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2048_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat2 @ B @ ( insert_nat2 @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_2049_leD,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ~ ( ord_less_assn @ X @ Y ) ) ).

% leD
thf(fact_2050_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_2051_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_2052_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_2053_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_2054_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_2055_nless__le,axiom,
    ! [A: assn,B: assn] :
      ( ( ~ ( ord_less_assn @ A @ B ) )
      = ( ~ ( ord_less_eq_assn @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2056_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2057_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2058_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2059_antisym__conv1,axiom,
    ! [X: assn,Y: assn] :
      ( ~ ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2060_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2061_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2062_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_2063_antisym__conv2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ~ ( ord_less_assn @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2064_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2065_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2066_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_2067_less__le__not__le,axiom,
    ( ord_less_assn
    = ( ^ [X3: assn,Y3: assn] :
          ( ( ord_less_eq_assn @ X3 @ Y3 )
          & ~ ( ord_less_eq_assn @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_2068_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_2069_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ~ ( ord_less_eq_int @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_2070_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y3 )
          & ~ ( ord_less_eq_set_nat @ Y3 @ X3 ) ) ) ) ).

% less_le_not_le
thf(fact_2071_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2072_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_2073_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_assn @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2074_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2075_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_int @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2076_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_set_nat @ A2 @ B2 )
          | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2077_order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2078_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2079_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2080_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2081_order_Ostrict__trans1,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2082_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2083_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2084_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_2085_order_Ostrict__trans2,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ord_less_assn @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2086_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2087_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2088_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_2089_order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ~ ( ord_less_eq_assn @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2090_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2091_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_less_eq_int @ A2 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2092_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( ord_less_eq_set_nat @ A2 @ B2 )
          & ~ ( ord_less_eq_set_nat @ B2 @ A2 ) ) ) ) ).

% order.strict_iff_not
thf(fact_2093_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( ord_less_assn @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2094_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2095_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_int @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2096_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_set_nat @ B2 @ A2 )
          | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_2097_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( ord_less_eq_assn @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2098_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2099_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2100_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_2101_dual__order_Ostrict__trans1,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_2102_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_2103_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_2104_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C2 @ B )
       => ( ord_less_set_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_2105_dual__order_Ostrict__trans2,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C2 @ B )
       => ( ord_less_assn @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_2106_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_2107_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_2108_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ B )
       => ( ord_less_set_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_2109_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( ord_less_eq_assn @ B2 @ A2 )
          & ~ ( ord_less_eq_assn @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2110_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2111_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_less_eq_int @ B2 @ A2 )
          & ~ ( ord_less_eq_int @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2112_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( ord_less_eq_set_nat @ B2 @ A2 )
          & ~ ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_2113_order_Ostrict__implies__order,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_2114_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_2115_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_2116_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_2117_dual__order_Ostrict__implies__order,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ord_less_eq_assn @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_2118_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_2119_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_2120_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_2121_order__le__less,axiom,
    ( ord_less_eq_assn
    = ( ^ [X3: assn,Y3: assn] :
          ( ( ord_less_assn @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_2122_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_2123_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_int @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_2124_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( ord_less_set_nat @ X3 @ Y3 )
          | ( X3 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_2125_order__less__le,axiom,
    ( ord_less_assn
    = ( ^ [X3: assn,Y3: assn] :
          ( ( ord_less_eq_assn @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_2126_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_2127_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X3: int,Y3: int] :
          ( ( ord_less_eq_int @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_2128_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X3 @ Y3 )
          & ( X3 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_2129_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2130_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_2131_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2132_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_2133_order__less__imp__le,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ord_less_eq_assn @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2134_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2135_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2136_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_2137_order__le__neq__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( A != B )
       => ( ord_less_assn @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_2138_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_2139_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_2140_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_2141_order__neq__le__trans,axiom,
    ! [A: assn,B: assn] :
      ( ( A != B )
     => ( ( ord_less_eq_assn @ A @ B )
       => ( ord_less_assn @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_2142_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_2143_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_2144_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_2145_order__le__less__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2146_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2147_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2148_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_le_less_trans
thf(fact_2149_order__less__le__trans,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ Y @ Z )
       => ( ord_less_assn @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2150_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2151_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2152_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z )
       => ( ord_less_set_nat @ X @ Z ) ) ) ).

% order_less_le_trans
thf(fact_2153_order__le__less__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2154_order__le__less__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2155_order__le__less__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_eq_assn @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2156_order__le__less__subst1,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2157_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2158_order__le__less__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2159_order__le__less__subst1,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2160_order__le__less__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2161_order__le__less__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2162_order__le__less__subst1,axiom,
    ! [A: set_nat,F: assn > set_nat,B: assn,C2: assn] :
      ( ( ord_less_eq_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_2163_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2164_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2165_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2166_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2167_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2168_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2169_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2170_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2171_order__le__less__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2172_order__le__less__subst2,axiom,
    ! [A: assn,B: assn,F: assn > set_nat,C2: set_nat] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_set_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_2173_order__less__le__subst1,axiom,
    ! [A: assn,F: assn > assn,B: assn,C2: assn] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2174_order__less__le__subst1,axiom,
    ! [A: nat,F: assn > nat,B: assn,C2: assn] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2175_order__less__le__subst1,axiom,
    ! [A: int,F: assn > int,B: assn,C2: assn] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2176_order__less__le__subst1,axiom,
    ! [A: assn,F: nat > assn,B: nat,C2: nat] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2177_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2178_order__less__le__subst1,axiom,
    ! [A: int,F: nat > int,B: nat,C2: nat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2179_order__less__le__subst1,axiom,
    ! [A: assn,F: int > assn,B: int,C2: int] :
      ( ( ord_less_assn @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2180_order__less__le__subst1,axiom,
    ! [A: nat,F: int > nat,B: int,C2: int] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2181_order__less__le__subst1,axiom,
    ! [A: int,F: int > int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ord_less_eq_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2182_order__less__le__subst1,axiom,
    ! [A: set_nat,F: assn > set_nat,B: assn,C2: assn] :
      ( ( ord_less_set_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_assn @ B @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ord_less_eq_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_2183_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > assn,C2: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2184_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > assn,C2: assn] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2185_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > assn,C2: assn] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_assn @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_assn @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_assn @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2186_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > nat,C2: nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2187_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2188_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > nat,C2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2189_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > int,C2: int] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2190_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > int,C2: int] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: nat,Y2: nat] :
              ( ( ord_less_nat @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2191_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
       => ( ! [X2: int,Y2: int] :
              ( ( ord_less_int @ X2 @ Y2 )
             => ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2192_order__less__le__subst2,axiom,
    ! [A: assn,B: assn,F: assn > set_nat,C2: set_nat] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_less_eq_set_nat @ ( F @ B ) @ C2 )
       => ( ! [X2: assn,Y2: assn] :
              ( ( ord_less_assn @ X2 @ Y2 )
             => ( ord_less_set_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
         => ( ord_less_set_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_2193_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_2194_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_2195_order__le__imp__less__or__eq,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_assn @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_2196_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_2197_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_2198_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_2199_Int__insert__right,axiom,
    ! [A: $o,A3: set_o,B3: set_o] :
      ( ( ( member_o @ A @ A3 )
       => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
          = ( insert_o @ A @ ( inf_inf_set_o @ A3 @ B3 ) ) ) )
      & ( ~ ( member_o @ A @ A3 )
       => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A @ B3 ) )
          = ( inf_inf_set_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_2200_Int__insert__right,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A @ A3 )
       => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
          = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A @ A3 )
       => ( ( inf_in1906310914598751387_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ B3 ) )
          = ( inf_in1906310914598751387_nat_o @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_2201_Int__insert__right,axiom,
    ! [A: nat,A3: set_nat,B3: set_nat] :
      ( ( ( member_nat2 @ A @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
          = ( insert_nat2 @ A @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) )
      & ( ~ ( member_nat2 @ A @ A3 )
       => ( ( inf_inf_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
          = ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_2202_Int__insert__right,axiom,
    ! [A: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A3 )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A @ B3 ) )
          = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_insert_right
thf(fact_2203_Int__insert__left,axiom,
    ! [A: $o,C3: set_o,B3: set_o] :
      ( ( ( member_o @ A @ C3 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C3 )
          = ( insert_o @ A @ ( inf_inf_set_o @ B3 @ C3 ) ) ) )
      & ( ~ ( member_o @ A @ C3 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B3 ) @ C3 )
          = ( inf_inf_set_o @ B3 @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_2204_Int__insert__left,axiom,
    ! [A: produc3658429121746597890et_nat > $o,C3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ A @ C3 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C3 )
          = ( insert5175938949040314269_nat_o @ A @ ( inf_in1906310914598751387_nat_o @ B3 @ C3 ) ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ A @ C3 )
       => ( ( inf_in1906310914598751387_nat_o @ ( insert5175938949040314269_nat_o @ A @ B3 ) @ C3 )
          = ( inf_in1906310914598751387_nat_o @ B3 @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_2205_Int__insert__left,axiom,
    ! [A: nat,C3: set_nat,B3: set_nat] :
      ( ( ( member_nat2 @ A @ C3 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C3 )
          = ( insert_nat2 @ A @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) )
      & ( ~ ( member_nat2 @ A @ C3 )
       => ( ( inf_inf_set_nat @ ( insert_nat2 @ A @ B3 ) @ C3 )
          = ( inf_inf_set_nat @ B3 @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_2206_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ C3 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C3 )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ C3 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B3 ) @ C3 )
          = ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ) ).

% Int_insert_left
thf(fact_2207_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2208_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2209_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_2210_bot_Oextremum__uniqueI,axiom,
    ! [A: assn] :
      ( ( ord_less_eq_assn @ A @ bot_bot_assn )
     => ( A = bot_bot_assn ) ) ).

% bot.extremum_uniqueI
thf(fact_2211_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2212_bot_Oextremum__uniqueI,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
     => ( A = bot_bot_set_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_2213_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_2214_bot_Oextremum__unique,axiom,
    ! [A: assn] :
      ( ( ord_less_eq_assn @ A @ bot_bot_assn )
      = ( A = bot_bot_assn ) ) ).

% bot.extremum_unique
thf(fact_2215_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_2216_bot_Oextremum__unique,axiom,
    ! [A: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
      = ( A = bot_bot_set_nat ) ) ).

% bot.extremum_unique
thf(fact_2217_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_2218_bot_Oextremum,axiom,
    ! [A: assn] : ( ord_less_eq_assn @ bot_bot_assn @ A ) ).

% bot.extremum
thf(fact_2219_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_2220_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_2221_inf_OcoboundedI2,axiom,
    ! [B: product_unit,C2: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ C2 )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2222_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2223_inf_OcoboundedI2,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2224_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2225_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ B @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2226_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_2227_inf_OcoboundedI1,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ C2 )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2228_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2229_inf_OcoboundedI1,axiom,
    ! [A: assn,C2: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ C2 )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2230_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2231_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2232_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_2233_inf_Oabsorb__iff2,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [B2: product_unit,A2: product_unit] :
          ( ( inf_inf_Product_unit @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2234_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2235_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( inf_inf_assn @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2236_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( inf_inf_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2237_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( inf_inf_int @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2238_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( inf_inf_set_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% inf.absorb_iff2
thf(fact_2239_inf_Oabsorb__iff1,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [A2: product_unit,B2: product_unit] :
          ( ( inf_inf_Product_unit @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2240_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2241_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( inf_inf_assn @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2242_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( inf_inf_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2243_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( inf_inf_int @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2244_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( inf_inf_set_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% inf.absorb_iff1
thf(fact_2245_inf_Ocobounded2,axiom,
    ! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2246_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2247_inf_Ocobounded2,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2248_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2249_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2250_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_2251_inf_Ocobounded1,axiom,
    ! [A: product_unit,B: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2252_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2253_inf_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2254_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2255_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2256_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_2257_inf_Oorder__iff,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [A2: product_unit,B2: product_unit] :
          ( A2
          = ( inf_inf_Product_unit @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2258_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
          ( A2
          = ( inf_in2572325071724192079at_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2259_inf_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( A2
          = ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2260_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( A2
          = ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2261_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( A2
          = ( inf_inf_int @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2262_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( A2
          = ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).

% inf.order_iff
thf(fact_2263_inf__greatest,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( ord_le3221252021190050221t_unit @ X @ Z )
       => ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2264_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X @ Z )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2265_inf__greatest,axiom,
    ! [X: assn,Y: assn,Z: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_less_eq_assn @ X @ Z )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2266_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2267_inf__greatest,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2268_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).

% inf_greatest
thf(fact_2269_inf_OboundedI,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( ( ord_le3221252021190050221t_unit @ A @ C2 )
       => ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2270_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C2 )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2271_inf_OboundedI,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( ord_less_eq_assn @ A @ C2 )
       => ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2272_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C2 )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2273_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C2 )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2274_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C2 )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_2275_inf_OboundedE,axiom,
    ! [A: product_unit,B: product_unit,C2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ ( inf_inf_Product_unit @ B @ C2 ) )
     => ~ ( ( ord_le3221252021190050221t_unit @ A @ B )
         => ~ ( ord_le3221252021190050221t_unit @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2276_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2277_inf_OboundedE,axiom,
    ! [A: assn,B: assn,C2: assn] :
      ( ( ord_less_eq_assn @ A @ ( inf_inf_assn @ B @ C2 ) )
     => ~ ( ( ord_less_eq_assn @ A @ B )
         => ~ ( ord_less_eq_assn @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2278_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2279_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2280_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_2281_inf__absorb2,axiom,
    ! [Y: product_unit,X: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ Y @ X )
     => ( ( inf_inf_Product_unit @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2282_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y @ X )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2283_inf__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( inf_inf_assn @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2284_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2285_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2286_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( inf_inf_set_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_2287_inf__absorb1,axiom,
    ! [X: product_unit,Y: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ Y )
     => ( ( inf_inf_Product_unit @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2288_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2289_inf__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( inf_inf_assn @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2290_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2291_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2292_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( inf_inf_set_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_2293_inf_Oabsorb2,axiom,
    ! [B: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ A )
     => ( ( inf_inf_Product_unit @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2294_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2295_inf_Oabsorb2,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( inf_inf_assn @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2296_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2297_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2298_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_2299_inf_Oabsorb1,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( ( inf_inf_Product_unit @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2300_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2301_inf_Oabsorb1,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( inf_inf_assn @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2302_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2303_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2304_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_2305_le__iff__inf,axiom,
    ( ord_le3221252021190050221t_unit
    = ( ^ [X3: product_unit,Y3: product_unit] :
          ( ( inf_inf_Product_unit @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2306_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2307_le__iff__inf,axiom,
    ( ord_less_eq_assn
    = ( ^ [X3: assn,Y3: assn] :
          ( ( inf_inf_assn @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2308_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( inf_inf_nat @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2309_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y3: int] :
          ( ( inf_inf_int @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2310_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( inf_inf_set_nat @ X3 @ Y3 )
          = X3 ) ) ) ).

% le_iff_inf
thf(fact_2311_inf__unique,axiom,
    ! [F: product_unit > product_unit > product_unit,X: product_unit,Y: product_unit] :
      ( ! [X2: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: product_unit,Y2: product_unit] : ( ord_le3221252021190050221t_unit @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: product_unit,Y2: product_unit,Z3: product_unit] :
              ( ( ord_le3221252021190050221t_unit @ X2 @ Y2 )
             => ( ( ord_le3221252021190050221t_unit @ X2 @ Z3 )
               => ( ord_le3221252021190050221t_unit @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_Product_unit @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2312_inf__unique,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X2 @ Y2 )
             => ( ( ord_le3146513528884898305at_nat @ X2 @ Z3 )
               => ( ord_le3146513528884898305at_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2313_inf__unique,axiom,
    ! [F: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : ( ord_less_eq_assn @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: assn,Y2: assn] : ( ord_less_eq_assn @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( ( ord_less_eq_assn @ X2 @ Y2 )
             => ( ( ord_less_eq_assn @ X2 @ Z3 )
               => ( ord_less_eq_assn @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_assn @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2314_inf__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y2 )
             => ( ( ord_less_eq_nat @ X2 @ Z3 )
               => ( ord_less_eq_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2315_inf__unique,axiom,
    ! [F: int > int > int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( ( ord_less_eq_int @ X2 @ Y2 )
             => ( ( ord_less_eq_int @ X2 @ Z3 )
               => ( ord_less_eq_int @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2316_inf__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ X2 )
     => ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y2 ) @ Y2 )
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ X2 @ Y2 )
             => ( ( ord_less_eq_set_nat @ X2 @ Z3 )
               => ( ord_less_eq_set_nat @ X2 @ ( F @ Y2 @ Z3 ) ) ) )
         => ( ( inf_inf_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_2317_inf_OorderI,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( A
        = ( inf_inf_Product_unit @ A @ B ) )
     => ( ord_le3221252021190050221t_unit @ A @ B ) ) ).

% inf.orderI
thf(fact_2318_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_2319_inf_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( inf_inf_assn @ A @ B ) )
     => ( ord_less_eq_assn @ A @ B ) ) ).

% inf.orderI
thf(fact_2320_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_2321_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_2322_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( inf_inf_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_2323_inf_OorderE,axiom,
    ! [A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ B )
     => ( A
        = ( inf_inf_Product_unit @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2324_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2325_inf_OorderE,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( A
        = ( inf_inf_assn @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2326_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2327_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2328_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( A
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_2329_le__infI2,axiom,
    ! [B: product_unit,X: product_unit,A: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ B @ X )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2330_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2331_le__infI2,axiom,
    ! [B: assn,X: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2332_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2333_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_eq_int @ B @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2334_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_2335_le__infI1,axiom,
    ! [A: product_unit,X: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ X )
     => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2336_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2337_le__infI1,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X )
     => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2338_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2339_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2340_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_2341_inf__mono,axiom,
    ! [A: product_unit,C2: product_unit,B: product_unit,D2: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ A @ C2 )
     => ( ( ord_le3221252021190050221t_unit @ B @ D2 )
       => ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ A @ B ) @ ( inf_inf_Product_unit @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2342_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2343_inf__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C2 )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( inf_inf_assn @ A @ B ) @ ( inf_inf_assn @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2344_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2345_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2346_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).

% inf_mono
thf(fact_2347_le__infI,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ A )
     => ( ( ord_le3221252021190050221t_unit @ X @ B )
       => ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2348_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ X @ B )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2349_le__infI,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ A )
     => ( ( ord_less_eq_assn @ X @ B )
       => ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2350_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2351_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ X @ B )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2352_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X @ B )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_2353_le__infE,axiom,
    ! [X: product_unit,A: product_unit,B: product_unit] :
      ( ( ord_le3221252021190050221t_unit @ X @ ( inf_inf_Product_unit @ A @ B ) )
     => ~ ( ( ord_le3221252021190050221t_unit @ X @ A )
         => ~ ( ord_le3221252021190050221t_unit @ X @ B ) ) ) ).

% le_infE
thf(fact_2354_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X @ A )
         => ~ ( ord_le3146513528884898305at_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_2355_le__infE,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ ( inf_inf_assn @ A @ B ) )
     => ~ ( ( ord_less_eq_assn @ X @ A )
         => ~ ( ord_less_eq_assn @ X @ B ) ) ) ).

% le_infE
thf(fact_2356_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_2357_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
     => ~ ( ( ord_less_eq_int @ X @ A )
         => ~ ( ord_less_eq_int @ X @ B ) ) ) ).

% le_infE
thf(fact_2358_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A )
         => ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_2359_inf__le2,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2360_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2361_inf__le2,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2362_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2363_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2364_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_2365_inf__le1,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2366_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2367_inf__le1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2368_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2369_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2370_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_2371_inf__sup__ord_I1_J,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2372_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2373_inf__sup__ord_I1_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2374_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2375_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2376_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_2377_inf__sup__ord_I2_J,axiom,
    ! [X: product_unit,Y: product_unit] : ( ord_le3221252021190050221t_unit @ ( inf_inf_Product_unit @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2378_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2379_inf__sup__ord_I2_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ ( inf_inf_assn @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2380_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2381_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2382_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_2383_inf__sup__ord_I4_J,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2384_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2385_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2386_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_2387_inf__sup__ord_I3_J,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2388_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2389_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2390_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_2391_le__supE,axiom,
    ! [A: assn,B: assn,X: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_assn @ A @ X )
         => ~ ( ord_less_eq_assn @ B @ X ) ) ) ).

% le_supE
thf(fact_2392_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_2393_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_int @ A @ X )
         => ~ ( ord_less_eq_int @ B @ X ) ) ) ).

% le_supE
thf(fact_2394_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A @ X )
         => ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_2395_le__supI,axiom,
    ! [A: assn,X: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ X )
     => ( ( ord_less_eq_assn @ B @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_2396_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_2397_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ( ord_less_eq_int @ B @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_2398_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ( ord_less_eq_set_nat @ B @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_2399_sup__ge1,axiom,
    ! [X: assn,Y: assn] : ( ord_less_eq_assn @ X @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge1
thf(fact_2400_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2401_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_2402_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_2403_sup__ge2,axiom,
    ! [Y: assn,X: assn] : ( ord_less_eq_assn @ Y @ ( sup_sup_assn @ X @ Y ) ) ).

% sup_ge2
thf(fact_2404_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2405_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_2406_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_2407_le__supI1,axiom,
    ! [X: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ X @ A )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI1
thf(fact_2408_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_2409_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_2410_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_2411_le__supI2,axiom,
    ! [X: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ X @ B )
     => ( ord_less_eq_assn @ X @ ( sup_sup_assn @ A @ B ) ) ) ).

% le_supI2
thf(fact_2412_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_2413_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X @ B )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_2414_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_2415_sup_Omono,axiom,
    ! [C2: assn,A: assn,D2: assn,B: assn] :
      ( ( ord_less_eq_assn @ C2 @ A )
     => ( ( ord_less_eq_assn @ D2 @ B )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ C2 @ D2 ) @ ( sup_sup_assn @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2416_sup_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D2 ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2417_sup_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D2 ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2418_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ( ord_less_eq_set_nat @ D2 @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D2 ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2419_sup__mono,axiom,
    ! [A: assn,C2: assn,B: assn,D2: assn] :
      ( ( ord_less_eq_assn @ A @ C2 )
     => ( ( ord_less_eq_assn @ B @ D2 )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ A @ B ) @ ( sup_sup_assn @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_2420_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_2421_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_2422_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).

% sup_mono
thf(fact_2423_sup__least,axiom,
    ! [Y: assn,X: assn,Z: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_less_eq_assn @ Z @ X )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2424_sup__least,axiom,
    ! [Y: nat,X: nat,Z: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2425_sup__least,axiom,
    ! [Y: int,X: int,Z: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2426_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).

% sup_least
thf(fact_2427_le__iff__sup,axiom,
    ( ord_less_eq_assn
    = ( ^ [X3: assn,Y3: assn] :
          ( ( sup_sup_assn @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2428_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X3: nat,Y3: nat] :
          ( ( sup_sup_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2429_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X3: int,Y3: int] :
          ( ( sup_sup_int @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2430_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X3: set_nat,Y3: set_nat] :
          ( ( sup_sup_set_nat @ X3 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2431_sup_OorderE,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( A
        = ( sup_sup_assn @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2432_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2433_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2434_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2435_sup_OorderI,axiom,
    ! [A: assn,B: assn] :
      ( ( A
        = ( sup_sup_assn @ A @ B ) )
     => ( ord_less_eq_assn @ B @ A ) ) ).

% sup.orderI
thf(fact_2436_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_2437_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_2438_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_2439_sup__unique,axiom,
    ! [F: assn > assn > assn,X: assn,Y: assn] :
      ( ! [X2: assn,Y2: assn] : ( ord_less_eq_assn @ X2 @ ( F @ X2 @ Y2 ) )
     => ( ! [X2: assn,Y2: assn] : ( ord_less_eq_assn @ Y2 @ ( F @ X2 @ Y2 ) )
       => ( ! [X2: assn,Y2: assn,Z3: assn] :
              ( ( ord_less_eq_assn @ Y2 @ X2 )
             => ( ( ord_less_eq_assn @ Z3 @ X2 )
               => ( ord_less_eq_assn @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_assn @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2440_sup__unique,axiom,
    ! [F: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y2 ) )
     => ( ! [X2: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X2 @ Y2 ) )
       => ( ! [X2: nat,Y2: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y2 @ X2 )
             => ( ( ord_less_eq_nat @ Z3 @ X2 )
               => ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2441_sup__unique,axiom,
    ! [F: int > int > int,X: int,Y: int] :
      ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ X2 @ ( F @ X2 @ Y2 ) )
     => ( ! [X2: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X2 @ Y2 ) )
       => ( ! [X2: int,Y2: int,Z3: int] :
              ( ( ord_less_eq_int @ Y2 @ X2 )
             => ( ( ord_less_eq_int @ Z3 @ X2 )
               => ( ord_less_eq_int @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2442_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y2 ) )
     => ( ! [X2: set_nat,Y2: set_nat] : ( ord_less_eq_set_nat @ Y2 @ ( F @ X2 @ Y2 ) )
       => ( ! [X2: set_nat,Y2: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ Y2 @ X2 )
             => ( ( ord_less_eq_set_nat @ Z3 @ X2 )
               => ( ord_less_eq_set_nat @ ( F @ Y2 @ Z3 ) @ X2 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2443_sup_Oabsorb1,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( sup_sup_assn @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2444_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2445_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2446_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2447_sup_Oabsorb2,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_eq_assn @ A @ B )
     => ( ( sup_sup_assn @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2448_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2449_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2450_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2451_sup__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( sup_sup_assn @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2452_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2453_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2454_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( sup_sup_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_2455_sup__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( sup_sup_assn @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2456_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2457_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2458_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( sup_sup_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2459_sup_OboundedE,axiom,
    ! [B: assn,C2: assn,A: assn] :
      ( ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_assn @ B @ A )
         => ~ ( ord_less_eq_assn @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_2460_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_2461_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_2462_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_2463_sup_OboundedI,axiom,
    ! [B: assn,A: assn,C2: assn] :
      ( ( ord_less_eq_assn @ B @ A )
     => ( ( ord_less_eq_assn @ C2 @ A )
       => ( ord_less_eq_assn @ ( sup_sup_assn @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2464_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2465_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2466_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2467_sup_Oorder__iff,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( A2
          = ( sup_sup_assn @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_2468_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_2469_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( sup_sup_int @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_2470_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( A2
          = ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% sup.order_iff
thf(fact_2471_sup_Ocobounded1,axiom,
    ! [A: assn,B: assn] : ( ord_less_eq_assn @ A @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2472_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2473_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2474_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2475_sup_Ocobounded2,axiom,
    ! [B: assn,A: assn] : ( ord_less_eq_assn @ B @ ( sup_sup_assn @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2476_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2477_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2478_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2479_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_assn
    = ( ^ [B2: assn,A2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_2480_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_2481_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_2482_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B2: set_nat,A2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% sup.absorb_iff1
thf(fact_2483_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( sup_sup_assn @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_2484_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( sup_sup_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_2485_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( sup_sup_int @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_2486_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] :
          ( ( sup_sup_set_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% sup.absorb_iff2
thf(fact_2487_sup_OcoboundedI1,axiom,
    ! [C2: assn,A: assn,B: assn] :
      ( ( ord_less_eq_assn @ C2 @ A )
     => ( ord_less_eq_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2488_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2489_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2490_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2491_sup_OcoboundedI2,axiom,
    ! [C2: assn,B: assn,A: assn] :
      ( ( ord_less_eq_assn @ C2 @ B )
     => ( ord_less_eq_assn @ C2 @ ( sup_sup_assn @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2492_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2493_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2494_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ B )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2495_Int__Collect__mono,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,P: ( produc3658429121746597890et_nat > $o ) > $o,Q: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ B3 )
     => ( ! [X2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ A3 )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le2965882846123202637_nat_o @ ( inf_in1906310914598751387_nat_o @ A3 @ ( collec939566748876313656_nat_o @ P ) ) @ ( inf_in1906310914598751387_nat_o @ B3 @ ( collec939566748876313656_nat_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2496_Int__Collect__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ! [X2: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X2 @ A3 )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B3 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2497_Int__Collect__mono,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
           => ( ( P @ X2 )
             => ( Q @ X2 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B3 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_2498_Int__greatest,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C3 @ A3 )
     => ( ( ord_le3146513528884898305at_nat @ C3 @ B3 )
       => ( ord_le3146513528884898305at_nat @ C3 @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_2499_Int__greatest,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ A3 )
     => ( ( ord_less_eq_set_nat @ C3 @ B3 )
       => ( ord_less_eq_set_nat @ C3 @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ).

% Int_greatest
thf(fact_2500_Int__absorb2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ B3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_2501_Int__absorb2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Int_absorb2
thf(fact_2502_Int__absorb1,axiom,
    ! [B3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B3 @ A3 )
     => ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_2503_Int__absorb1,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( inf_inf_set_nat @ A3 @ B3 )
        = B3 ) ) ).

% Int_absorb1
thf(fact_2504_Int__lower2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_2505_Int__lower2,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ B3 ) ).

% Int_lower2
thf(fact_2506_Int__lower1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_2507_Int__lower1,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ A3 ) ).

% Int_lower1
thf(fact_2508_Int__mono,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A3 @ C3 )
     => ( ( ord_le3146513528884898305at_nat @ B3 @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ D ) ) ) ) ).

% Int_mono
thf(fact_2509_Int__mono,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ C3 @ D ) ) ) ) ).

% Int_mono
thf(fact_2510_inter__eq__subsetI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S @ S3 )
     => ( ( ( inf_in2572325071724192079at_nat @ A3 @ S3 )
          = ( inf_in2572325071724192079at_nat @ B3 @ S3 ) )
       => ( ( inf_in2572325071724192079at_nat @ A3 @ S )
          = ( inf_in2572325071724192079at_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2511_inter__eq__subsetI,axiom,
    ! [S: set_nat,S3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ S @ S3 )
     => ( ( ( inf_inf_set_nat @ A3 @ S3 )
          = ( inf_inf_set_nat @ B3 @ S3 ) )
       => ( ( inf_inf_set_nat @ A3 @ S )
          = ( inf_inf_set_nat @ B3 @ S ) ) ) ) ).

% inter_eq_subsetI
thf(fact_2512_Un__mono,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ ( sup_sup_set_nat @ C3 @ D ) ) ) ) ).

% Un_mono
thf(fact_2513_Un__least,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 ) ) ) ).

% Un_least
thf(fact_2514_Un__upper1,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_upper1
thf(fact_2515_Un__upper2,axiom,
    ! [B3: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_upper2
thf(fact_2516_Un__absorb1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( sup_sup_set_nat @ A3 @ B3 )
        = B3 ) ) ).

% Un_absorb1
thf(fact_2517_Un__absorb2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( ord_less_eq_set_nat @ B3 @ A3 )
     => ( ( sup_sup_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Un_absorb2
thf(fact_2518_subset__UnE,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ C3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ~ ! [A6: set_nat] :
            ( ( ord_less_eq_set_nat @ A6 @ A3 )
           => ! [B7: set_nat] :
                ( ( ord_less_eq_set_nat @ B7 @ B3 )
               => ( C3
                 != ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ) ).

% subset_UnE
thf(fact_2519_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% subset_Un_eq
thf(fact_2520_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_2521_subset__psubset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_set_nat @ B3 @ C3 )
       => ( ord_less_set_nat @ A3 @ C3 ) ) ) ).

% subset_psubset_trans
thf(fact_2522_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_2523_psubset__subset__trans,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ord_less_set_nat @ A3 @ C3 ) ) ) ).

% psubset_subset_trans
thf(fact_2524_psubset__imp__subset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_2525_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% psubset_eq
thf(fact_2526_psubsetE,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ~ ( ( ord_less_eq_set_nat @ A3 @ B3 )
         => ( ord_less_eq_set_nat @ B3 @ A3 ) ) ) ).

% psubsetE
thf(fact_2527_less__eq__assn__def,axiom,
    ( ord_less_eq_assn
    = ( ^ [A2: assn,B2: assn] :
          ( A2
          = ( inf_inf_assn @ A2 @ B2 ) ) ) ) ).

% less_eq_assn_def
thf(fact_2528_Max__mono,axiom,
    ! [M: set_o,N4: set_o] :
      ( ( ord_less_eq_set_o @ M @ N4 )
     => ( ( M != bot_bot_set_o )
       => ( ( finite_finite_o @ N4 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ M ) @ ( lattic1921953407002678535_Max_o @ N4 ) ) ) ) ) ).

% Max_mono
thf(fact_2529_Max__mono,axiom,
    ! [M: set_int,N4: set_int] :
      ( ( ord_less_eq_set_int @ M @ N4 )
     => ( ( M != bot_bot_set_int )
       => ( ( finite_finite_int @ N4 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ M ) @ ( lattic8263393255366662781ax_int @ N4 ) ) ) ) ) ).

% Max_mono
thf(fact_2530_Max__mono,axiom,
    ! [M: set_nat,N4: set_nat] :
      ( ( ord_less_eq_set_nat @ M @ N4 )
     => ( ( M != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N4 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ M ) @ ( lattic8265883725875713057ax_nat @ N4 ) ) ) ) ) ).

% Max_mono
thf(fact_2531_Max_Osubset__imp,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ ( lattic1921953407002678535_Max_o @ B3 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2532_Max_Osubset__imp,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ ( lattic8263393255366662781ax_int @ B3 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2533_Max_Osubset__imp,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ).

% Max.subset_imp
thf(fact_2534_Min__antimono,axiom,
    ! [M: set_o,N4: set_o] :
      ( ( ord_less_eq_set_o @ M @ N4 )
     => ( ( M != bot_bot_set_o )
       => ( ( finite_finite_o @ N4 )
         => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ N4 ) @ ( lattic1973801136483472281_Min_o @ M ) ) ) ) ) ).

% Min_antimono
thf(fact_2535_Min__antimono,axiom,
    ! [M: set_int,N4: set_int] :
      ( ( ord_less_eq_set_int @ M @ N4 )
     => ( ( M != bot_bot_set_int )
       => ( ( finite_finite_int @ N4 )
         => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ N4 ) @ ( lattic8718645017227715691in_int @ M ) ) ) ) ) ).

% Min_antimono
thf(fact_2536_Min__antimono,axiom,
    ! [M: set_nat,N4: set_nat] :
      ( ( ord_less_eq_set_nat @ M @ N4 )
     => ( ( M != bot_bot_set_nat )
       => ( ( finite_finite_nat @ N4 )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ N4 ) @ ( lattic8721135487736765967in_nat @ M ) ) ) ) ) ).

% Min_antimono
thf(fact_2537_Min_Osubset__imp,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ B3 ) @ ( lattic1973801136483472281_Min_o @ A3 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2538_Min_Osubset__imp,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ B3 ) @ ( lattic8718645017227715691in_int @ A3 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2539_Min_Osubset__imp,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ B3 ) @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).

% Min.subset_imp
thf(fact_2540_Inf__fin_Osubset__imp,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ord_less_eq_o @ ( lattic4107685809792843317_fin_o @ B3 ) @ ( lattic4107685809792843317_fin_o @ A3 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_2541_Inf__fin_Osubset__imp,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( ord_less_eq_set_assn @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B3 )
         => ( ord_less_eq_assn @ ( lattic47131356835913163n_assn @ B3 ) @ ( lattic47131356835913163n_assn @ A3 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_2542_Inf__fin_Osubset__imp,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_int @ ( lattic5235898064620869839in_int @ B3 ) @ ( lattic5235898064620869839in_int @ A3 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_2543_Inf__fin_Osubset__imp,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B3 )
         => ( ord_less_eq_set_nat @ ( lattic3014633134055518761et_nat @ B3 ) @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_2544_Inf__fin_Osubset__imp,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic5238388535129920115in_nat @ B3 ) @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).

% Inf_fin.subset_imp
thf(fact_2545_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ ( lattic1508158080041050831_fin_o @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_2546_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_assn,B3: set_assn] :
      ( ( ord_less_eq_set_assn @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( finite_finite_assn @ B3 )
         => ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ ( lattic2150320897289308081n_assn @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_2547_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( finite_finite_int @ B3 )
         => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A3 ) @ ( lattic1091506334969745077in_int @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_2548_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_set_nat,B3: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( finite1152437895449049373et_nat @ B3 )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ ( lattic3835124923745554447et_nat @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_2549_Sup__fin_Osubset__imp,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ ( lattic1093996805478795353in_nat @ B3 ) ) ) ) ) ).

% Sup_fin.subset_imp
thf(fact_2550_infinite__finite__induct,axiom,
    ! [P: set_Pr4532377907799695533_nat_o > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ! [A7: set_Pr4532377907799695533_nat_o] :
          ( ~ ( finite3252695134891459830_nat_o @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,F4: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ F4 )
             => ( ~ ( member6576561426505652726_nat_o @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X2 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2551_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A3: set_o] :
      ( ! [A7: set_o] :
          ( ~ ( finite_finite_o @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X2: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X2 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2552_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A3: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat2 @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat2 @ X2 @ F4 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% infinite_finite_induct
thf(fact_2553_finite__ne__induct,axiom,
    ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F2 )
     => ( ( F2 != bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o] : ( P @ ( insert5175938949040314269_nat_o @ X2 @ bot_bo7824918357723582233_nat_o ) )
         => ( ! [X2: produc3658429121746597890et_nat > $o,F4: set_Pr4532377907799695533_nat_o] :
                ( ( finite3252695134891459830_nat_o @ F4 )
               => ( ( F4 != bot_bo7824918357723582233_nat_o )
                 => ( ~ ( member6576561426505652726_nat_o @ X2 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert5175938949040314269_nat_o @ X2 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2554_finite__ne__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( F2 != bot_bot_set_o )
       => ( ! [X2: $o] : ( P @ ( insert_o @ X2 @ bot_bot_set_o ) )
         => ( ! [X2: $o,F4: set_o] :
                ( ( finite_finite_o @ F4 )
               => ( ( F4 != bot_bot_set_o )
                 => ( ~ ( member_o @ X2 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_o @ X2 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2555_finite__ne__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( F2 != bot_bot_set_nat )
       => ( ! [X2: nat] : ( P @ ( insert_nat2 @ X2 @ bot_bot_set_nat ) )
         => ( ! [X2: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat2 @ X2 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat2 @ X2 @ F4 ) ) ) ) ) )
           => ( P @ F2 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_2556_finite__induct,axiom,
    ! [F2: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ F2 )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,F4: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ F4 )
             => ( ~ ( member6576561426505652726_nat_o @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert5175938949040314269_nat_o @ X2 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2557_finite__induct,axiom,
    ! [F2: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F2 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X2: $o,F4: set_o] :
              ( ( finite_finite_o @ F4 )
             => ( ~ ( member_o @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_o @ X2 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2558_finite__induct,axiom,
    ! [F2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X2: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat2 @ X2 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat2 @ X2 @ F4 ) ) ) ) )
         => ( P @ F2 ) ) ) ) ).

% finite_induct
thf(fact_2559_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A2: set_o] :
          ( ( A2 = bot_bot_set_o )
          | ? [A5: set_o,B2: $o] :
              ( ( A2
                = ( insert_o @ B2 @ A5 ) )
              & ( finite_finite_o @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_2560_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A2: set_nat] :
          ( ( A2 = bot_bot_set_nat )
          | ? [A5: set_nat,B2: nat] :
              ( ( A2
                = ( insert_nat2 @ B2 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_2561_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A7: set_o] :
              ( ? [A4: $o] :
                  ( A
                  = ( insert_o @ A4 @ A7 ) )
             => ~ ( finite_finite_o @ A7 ) ) ) ) ).

% finite.cases
thf(fact_2562_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A4: nat] :
                  ( A
                  = ( insert_nat2 @ A4 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_2563_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A2: $o] : ( sup_sup_set_o @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_2564_insert__is__Un,axiom,
    ( insert_nat2
    = ( ^ [A2: nat] : ( sup_sup_set_nat @ ( insert_nat2 @ A2 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_2565_Un__singleton__iff,axiom,
    ! [A3: set_o,B3: set_o,X: $o] :
      ( ( ( sup_sup_set_o @ A3 @ B3 )
        = ( insert_o @ X @ bot_bot_set_o ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B3
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B3 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B3
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_2566_Un__singleton__iff,axiom,
    ! [A3: set_nat,B3: set_nat,X: nat] :
      ( ( ( sup_sup_set_nat @ A3 @ B3 )
        = ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_2567_singleton__Un__iff,axiom,
    ! [X: $o,A3: set_o,B3: set_o] :
      ( ( ( insert_o @ X @ bot_bot_set_o )
        = ( sup_sup_set_o @ A3 @ B3 ) )
      = ( ( ( A3 = bot_bot_set_o )
          & ( B3
            = ( insert_o @ X @ bot_bot_set_o ) ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B3 = bot_bot_set_o ) )
        | ( ( A3
            = ( insert_o @ X @ bot_bot_set_o ) )
          & ( B3
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_2568_singleton__Un__iff,axiom,
    ! [X: nat,A3: set_nat,B3: set_nat] :
      ( ( ( insert_nat2 @ X @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( ( ( A3 = bot_bot_set_nat )
          & ( B3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        | ( ( A3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B3 = bot_bot_set_nat ) )
        | ( ( A3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) )
          & ( B3
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_2569_finite__has__minimal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X2: $o] :
            ( ( member_o @ X2 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2570_finite__has__minimal,axiom,
    ! [A3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ? [X2: assn] :
            ( ( member_assn @ X2 @ A3 )
            & ! [Xa2: assn] :
                ( ( member_assn @ Xa2 @ A3 )
               => ( ( ord_less_eq_assn @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2571_finite__has__minimal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat2 @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2572_finite__has__minimal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X2: int] :
            ( ( member_int2 @ X2 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2573_finite__has__minimal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ Xa2 @ X2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2574_finite__has__maximal,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ? [X2: $o] :
            ( ( member_o @ X2 @ A3 )
            & ! [Xa2: $o] :
                ( ( member_o @ Xa2 @ A3 )
               => ( ( ord_less_eq_o @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2575_finite__has__maximal,axiom,
    ! [A3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ? [X2: assn] :
            ( ( member_assn @ X2 @ A3 )
            & ! [Xa2: assn] :
                ( ( member_assn @ Xa2 @ A3 )
               => ( ( ord_less_eq_assn @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2576_finite__has__maximal,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ? [X2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
            & ! [Xa2: nat] :
                ( ( member_nat2 @ Xa2 @ A3 )
               => ( ( ord_less_eq_nat @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2577_finite__has__maximal,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ? [X2: int] :
            ( ( member_int2 @ X2 @ A3 )
            & ! [Xa2: int] :
                ( ( member_int2 @ Xa2 @ A3 )
               => ( ( ord_less_eq_int @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2578_finite__has__maximal,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ? [X2: set_nat] :
            ( ( member_set_nat @ X2 @ A3 )
            & ! [Xa2: set_nat] :
                ( ( member_set_nat @ Xa2 @ A3 )
               => ( ( ord_less_eq_set_nat @ X2 @ Xa2 )
                 => ( X2 = Xa2 ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2579_distrib__inf__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ ( inf_inf_Product_unit @ X @ Y ) @ ( inf_inf_Product_unit @ X @ Z ) ) @ ( inf_inf_Product_unit @ X @ ( sup_sup_Product_unit @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2580_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2581_distrib__inf__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ ( inf_inf_assn @ X @ Y ) @ ( inf_inf_assn @ X @ Z ) ) @ ( inf_inf_assn @ X @ ( sup_sup_assn @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2582_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2583_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2584_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).

% distrib_inf_le
thf(fact_2585_distrib__sup__le,axiom,
    ! [X: product_unit,Y: product_unit,Z: product_unit] : ( ord_le3221252021190050221t_unit @ ( sup_sup_Product_unit @ X @ ( inf_inf_Product_unit @ Y @ Z ) ) @ ( inf_inf_Product_unit @ ( sup_sup_Product_unit @ X @ Y ) @ ( sup_sup_Product_unit @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2586_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2587_distrib__sup__le,axiom,
    ! [X: assn,Y: assn,Z: assn] : ( ord_less_eq_assn @ ( sup_sup_assn @ X @ ( inf_inf_assn @ Y @ Z ) ) @ ( inf_inf_assn @ ( sup_sup_assn @ X @ Y ) @ ( sup_sup_assn @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2588_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2589_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2590_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).

% distrib_sup_le
thf(fact_2591_disjoint__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,A8: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,B8: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ A8 )
     => ( ( ord_le3146513528884898305at_nat @ B @ B8 )
       => ( ( ( inf_in2572325071724192079at_nat @ A8 @ B8 )
            = bot_bo2099793752762293965at_nat )
         => ( ( inf_in2572325071724192079at_nat @ A @ B )
            = bot_bo2099793752762293965at_nat ) ) ) ) ).

% disjoint_mono
thf(fact_2592_disjoint__mono,axiom,
    ! [A: set_o,A8: set_o,B: set_o,B8: set_o] :
      ( ( ord_less_eq_set_o @ A @ A8 )
     => ( ( ord_less_eq_set_o @ B @ B8 )
       => ( ( ( inf_inf_set_o @ A8 @ B8 )
            = bot_bot_set_o )
         => ( ( inf_inf_set_o @ A @ B )
            = bot_bot_set_o ) ) ) ) ).

% disjoint_mono
thf(fact_2593_disjoint__mono,axiom,
    ! [A: set_nat,A8: set_nat,B: set_nat,B8: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ A8 )
     => ( ( ord_less_eq_set_nat @ B @ B8 )
       => ( ( ( inf_inf_set_nat @ A8 @ B8 )
            = bot_bot_set_nat )
         => ( ( inf_inf_set_nat @ A @ B )
            = bot_bot_set_nat ) ) ) ) ).

% disjoint_mono
thf(fact_2594_Un__Int__assoc__eq,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
        = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) ) )
      = ( ord_le3146513528884898305at_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_2595_Un__Int__assoc__eq,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
        = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) )
      = ( ord_less_eq_set_nat @ C3 @ A3 ) ) ).

% Un_Int_assoc_eq
thf(fact_2596_finite__linorder__max__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A7 )
                   => ( ord_less_o @ X6 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2597_finite__linorder__max__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat2 @ X6 @ A7 )
                   => ( ord_less_nat @ X6 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat2 @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2598_finite__linorder__max__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X6: int] :
                    ( ( member_int2 @ X6 @ A7 )
                   => ( ord_less_int @ X6 @ B5 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int2 @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_2599_finite__linorder__min__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B5: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ! [X6: $o] :
                    ( ( member_o @ X6 @ A7 )
                   => ( ord_less_o @ B5 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_o @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2600_finite__linorder__min__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B5: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X6: nat] :
                    ( ( member_nat2 @ X6 @ A7 )
                   => ( ord_less_nat @ B5 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat2 @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2601_finite__linorder__min__induct,axiom,
    ! [A3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B5: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X6: int] :
                    ( ( member_int2 @ X6 @ A7 )
                   => ( ord_less_int @ B5 @ X6 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int2 @ B5 @ A7 ) ) ) ) )
         => ( P @ A3 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_2602_ShiftD,axiom,
    ! [Kl: list_nat,Kl2: set_list_nat,K: nat] :
      ( ( member_list_nat @ Kl @ ( bNF_Gr1872714664788909425ft_nat @ Kl2 @ K ) )
     => ( member_list_nat @ ( cons_nat @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_2603_ShiftD,axiom,
    ! [Kl: list_int,Kl2: set_list_int,K: int] :
      ( ( member_list_int @ Kl @ ( bNF_Gr1870224194279859149ft_int @ Kl2 @ K ) )
     => ( member_list_int @ ( cons_int @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_2604_ShiftD,axiom,
    ! [Kl: list_P8527749157015355191n_assn,Kl2: set_li5131720305576846103n_assn,K: produc6575502325842934193n_assn] :
      ( ( member852475432509897056n_assn @ Kl @ ( bNF_Gr4113829767105464016n_assn @ Kl2 @ K ) )
     => ( member852475432509897056n_assn @ ( cons_P2971678138204555879n_assn @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_2605_Max_OboundedI,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( ord_less_eq_o @ A4 @ X ) )
         => ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2606_Max_OboundedI,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( ( member_nat2 @ A4 @ A3 )
             => ( ord_less_eq_nat @ A4 @ X ) )
         => ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2607_Max_OboundedI,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( ( member_int2 @ A4 @ A3 )
             => ( ord_less_eq_int @ A4 @ X ) )
         => ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X ) ) ) ) ).

% Max.boundedI
thf(fact_2608_Max_OboundedE,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ X )
         => ! [A9: $o] :
              ( ( member_o @ A9 @ A3 )
             => ( ord_less_eq_o @ A9 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2609_Max_OboundedE,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ X )
         => ! [A9: nat] :
              ( ( member_nat2 @ A9 @ A3 )
             => ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2610_Max_OboundedE,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A3 ) @ X )
         => ! [A9: int] :
              ( ( member_int2 @ A9 @ A3 )
             => ( ord_less_eq_int @ A9 @ X ) ) ) ) ) ).

% Max.boundedE
thf(fact_2611_eq__Max__iff,axiom,
    ! [A3: set_o,M2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( M2
            = ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ( member_o @ M2 @ A3 )
            & ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X3 @ M2 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2612_eq__Max__iff,axiom,
    ! [A3: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( M2
            = ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ( member_nat2 @ M2 @ A3 )
            & ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2613_eq__Max__iff,axiom,
    ! [A3: set_int,M2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( M2
            = ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ( member_int2 @ M2 @ A3 )
            & ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X3 @ M2 ) ) ) ) ) ) ).

% eq_Max_iff
thf(fact_2614_Max__ge__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1921953407002678535_Max_o @ A3 ) )
          = ( ? [X3: $o] :
                ( ( member_o @ X3 @ A3 )
                & ( ord_less_eq_o @ X @ X3 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2615_Max__ge__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) )
          = ( ? [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
                & ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2616_Max__ge__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8263393255366662781ax_int @ A3 ) )
          = ( ? [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
                & ( ord_less_eq_int @ X @ X3 ) ) ) ) ) ) ).

% Max_ge_iff
thf(fact_2617_Max__eq__iff,axiom,
    ! [A3: set_o,M2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ( lattic1921953407002678535_Max_o @ A3 )
            = M2 )
          = ( ( member_o @ M2 @ A3 )
            & ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X3 @ M2 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2618_Max__eq__iff,axiom,
    ! [A3: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ( lattic8265883725875713057ax_nat @ A3 )
            = M2 )
          = ( ( member_nat2 @ M2 @ A3 )
            & ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2619_Max__eq__iff,axiom,
    ! [A3: set_int,M2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ( lattic8263393255366662781ax_int @ A3 )
            = M2 )
          = ( ( member_int2 @ M2 @ A3 )
            & ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X3 @ M2 ) ) ) ) ) ) ).

% Max_eq_iff
thf(fact_2620_Min_OboundedI,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( ord_less_eq_o @ X @ A4 ) )
         => ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2621_Min_OboundedI,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( ( member_nat2 @ A4 @ A3 )
             => ( ord_less_eq_nat @ X @ A4 ) )
         => ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2622_Min_OboundedI,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( ( member_int2 @ A4 @ A3 )
             => ( ord_less_eq_int @ X @ A4 ) )
         => ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A3 ) ) ) ) ) ).

% Min.boundedI
thf(fact_2623_Min_OboundedE,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) )
         => ! [A9: $o] :
              ( ( member_o @ A9 @ A3 )
             => ( ord_less_eq_o @ X @ A9 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2624_Min_OboundedE,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) )
         => ! [A9: nat] :
              ( ( member_nat2 @ A9 @ A3 )
             => ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2625_Min_OboundedE,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A3 ) )
         => ! [A9: int] :
              ( ( member_int2 @ A9 @ A3 )
             => ( ord_less_eq_int @ X @ A9 ) ) ) ) ) ).

% Min.boundedE
thf(fact_2626_eq__Min__iff,axiom,
    ! [A3: set_o,M2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( M2
            = ( lattic1973801136483472281_Min_o @ A3 ) )
          = ( ( member_o @ M2 @ A3 )
            & ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ M2 @ X3 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2627_eq__Min__iff,axiom,
    ! [A3: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( M2
            = ( lattic8721135487736765967in_nat @ A3 ) )
          = ( ( member_nat2 @ M2 @ A3 )
            & ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ M2 @ X3 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2628_eq__Min__iff,axiom,
    ! [A3: set_int,M2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( M2
            = ( lattic8718645017227715691in_int @ A3 ) )
          = ( ( member_int2 @ M2 @ A3 )
            & ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ M2 @ X3 ) ) ) ) ) ) ).

% eq_Min_iff
thf(fact_2629_Min__le__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ X )
          = ( ? [X3: $o] :
                ( ( member_o @ X3 @ A3 )
                & ( ord_less_eq_o @ X3 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2630_Min__le__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ X )
          = ( ? [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
                & ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2631_Min__le__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic8718645017227715691in_int @ A3 ) @ X )
          = ( ? [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
                & ( ord_less_eq_int @ X3 @ X ) ) ) ) ) ) ).

% Min_le_iff
thf(fact_2632_Min__eq__iff,axiom,
    ! [A3: set_o,M2: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ( lattic1973801136483472281_Min_o @ A3 )
            = M2 )
          = ( ( member_o @ M2 @ A3 )
            & ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ M2 @ X3 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2633_Min__eq__iff,axiom,
    ! [A3: set_nat,M2: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ( lattic8721135487736765967in_nat @ A3 )
            = M2 )
          = ( ( member_nat2 @ M2 @ A3 )
            & ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ M2 @ X3 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2634_Min__eq__iff,axiom,
    ! [A3: set_int,M2: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ( lattic8718645017227715691in_int @ A3 )
            = M2 )
          = ( ( member_int2 @ M2 @ A3 )
            & ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ M2 @ X3 ) ) ) ) ) ) ).

% Min_eq_iff
thf(fact_2635_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ X )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_2636_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ X )
          = ( ! [X3: assn] :
                ( ( member_assn @ X3 @ A3 )
               => ( ord_less_eq_assn @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_2637_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_2638_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A3 ) @ X )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_2639_Sup__fin_Obounded__iff,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X )
          = ( ! [X3: set_nat] :
                ( ( member_set_nat @ X3 @ A3 )
               => ( ord_less_eq_set_nat @ X3 @ X ) ) ) ) ) ) ).

% Sup_fin.bounded_iff
thf(fact_2640_Inf__fin_Obounded__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A3 ) )
          = ( ! [X3: $o] :
                ( ( member_o @ X3 @ A3 )
               => ( ord_less_eq_o @ X @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_2641_Inf__fin_Obounded__iff,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) )
          = ( ! [X3: assn] :
                ( ( member_assn @ X3 @ A3 )
               => ( ord_less_eq_assn @ X @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_2642_Inf__fin_Obounded__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
          = ( ! [X3: nat] :
                ( ( member_nat2 @ X3 @ A3 )
               => ( ord_less_eq_nat @ X @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_2643_Inf__fin_Obounded__iff,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A3 ) )
          = ( ! [X3: int] :
                ( ( member_int2 @ X3 @ A3 )
               => ( ord_less_eq_int @ X @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_2644_Inf__fin_Obounded__iff,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) )
          = ( ! [X3: set_nat] :
                ( ( member_set_nat @ X3 @ A3 )
               => ( ord_less_eq_set_nat @ X @ X3 ) ) ) ) ) ) ).

% Inf_fin.bounded_iff
thf(fact_2645_Sup__fin_OboundedI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ! [A4: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A4 @ A3 )
             => ( ord_le729326519192465773_nat_o @ A4 @ X ) )
         => ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2646_Sup__fin_OboundedI,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( ord_less_eq_o @ A4 @ X ) )
         => ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2647_Sup__fin_OboundedI,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ! [A4: assn] :
              ( ( member_assn @ A4 @ A3 )
             => ( ord_less_eq_assn @ A4 @ X ) )
         => ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2648_Sup__fin_OboundedI,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( ( member_nat2 @ A4 @ A3 )
             => ( ord_less_eq_nat @ A4 @ X ) )
         => ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2649_Sup__fin_OboundedI,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( ( member_int2 @ A4 @ A3 )
             => ( ord_less_eq_int @ A4 @ X ) )
         => ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2650_Sup__fin_OboundedI,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [A4: set_nat] :
              ( ( member_set_nat @ A4 @ A3 )
             => ( ord_less_eq_set_nat @ A4 @ X ) )
         => ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X ) ) ) ) ).

% Sup_fin.boundedI
thf(fact_2651_Sup__fin_OboundedE,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ( ord_le729326519192465773_nat_o @ ( lattic7320199455484906628_nat_o @ A3 ) @ X )
         => ! [A9: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A9 @ A3 )
             => ( ord_le729326519192465773_nat_o @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2652_Sup__fin_OboundedE,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ ( lattic1508158080041050831_fin_o @ A3 ) @ X )
         => ! [A9: $o] :
              ( ( member_o @ A9 @ A3 )
             => ( ord_less_eq_o @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2653_Sup__fin_OboundedE,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ ( lattic2150320897289308081n_assn @ A3 ) @ X )
         => ! [A9: assn] :
              ( ( member_assn @ A9 @ A3 )
             => ( ord_less_eq_assn @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2654_Sup__fin_OboundedE,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A3 ) @ X )
         => ! [A9: nat] :
              ( ( member_nat2 @ A9 @ A3 )
             => ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2655_Sup__fin_OboundedE,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A3 ) @ X )
         => ! [A9: int] :
              ( ( member_int2 @ A9 @ A3 )
             => ( ord_less_eq_int @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2656_Sup__fin_OboundedE,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ ( lattic3835124923745554447et_nat @ A3 ) @ X )
         => ! [A9: set_nat] :
              ( ( member_set_nat @ A9 @ A3 )
             => ( ord_less_eq_set_nat @ A9 @ X ) ) ) ) ) ).

% Sup_fin.boundedE
thf(fact_2657_Inf__fin_OboundedI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ! [A4: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A4 @ A3 )
             => ( ord_le729326519192465773_nat_o @ X @ A4 ) )
         => ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2658_Inf__fin_OboundedI,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [A4: $o] :
              ( ( member_o @ A4 @ A3 )
             => ( ord_less_eq_o @ X @ A4 ) )
         => ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2659_Inf__fin_OboundedI,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ! [A4: assn] :
              ( ( member_assn @ A4 @ A3 )
             => ( ord_less_eq_assn @ X @ A4 ) )
         => ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2660_Inf__fin_OboundedI,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [A4: nat] :
              ( ( member_nat2 @ A4 @ A3 )
             => ( ord_less_eq_nat @ X @ A4 ) )
         => ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2661_Inf__fin_OboundedI,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ! [A4: int] :
              ( ( member_int2 @ A4 @ A3 )
             => ( ord_less_eq_int @ X @ A4 ) )
         => ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2662_Inf__fin_OboundedI,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [A4: set_nat] :
              ( ( member_set_nat @ A4 @ A3 )
             => ( ord_less_eq_set_nat @ X @ A4 ) )
         => ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ).

% Inf_fin.boundedI
thf(fact_2663_Inf__fin_OboundedE,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ( ord_le729326519192465773_nat_o @ X @ ( lattic956194824204696298_nat_o @ A3 ) )
         => ! [A9: produc3658429121746597890et_nat > $o] :
              ( ( member6576561426505652726_nat_o @ A9 @ A3 )
             => ( ord_le729326519192465773_nat_o @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2664_Inf__fin_OboundedE,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( ord_less_eq_o @ X @ ( lattic4107685809792843317_fin_o @ A3 ) )
         => ! [A9: $o] :
              ( ( member_o @ A9 @ A3 )
             => ( ord_less_eq_o @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2665_Inf__fin_OboundedE,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ( ord_less_eq_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) )
         => ! [A9: assn] :
              ( ( member_assn @ A9 @ A3 )
             => ( ord_less_eq_assn @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2666_Inf__fin_OboundedE,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) )
         => ! [A9: nat] :
              ( ( member_nat2 @ A9 @ A3 )
             => ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2667_Inf__fin_OboundedE,axiom,
    ! [A3: set_int,X: int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A3 ) )
         => ! [A9: int] :
              ( ( member_int2 @ A9 @ A3 )
             => ( ord_less_eq_int @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2668_Inf__fin_OboundedE,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ( ord_less_eq_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) )
         => ! [A9: set_nat] :
              ( ( member_set_nat @ A9 @ A3 )
             => ( ord_less_eq_set_nat @ X @ A9 ) ) ) ) ) ).

% Inf_fin.boundedE
thf(fact_2669_arg__min__least,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,F: ( produc3658429121746597890et_nat > $o ) > nat] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( S != bot_bo7824918357723582233_nat_o )
       => ( ( member6576561426505652726_nat_o @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic4812295286663557444_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2670_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: $o > nat] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic2775856028456453135_o_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2671_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > nat] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat2 @ Y @ S )
         => ( ord_less_eq_nat @ ( F @ ( lattic7446932960582359483at_nat @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2672_arg__min__least,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,Y: produc3658429121746597890et_nat > $o,F: ( produc3658429121746597890et_nat > $o ) > int] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( S != bot_bo7824918357723582233_nat_o )
       => ( ( member6576561426505652726_nat_o @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic4809804816154507168_o_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2673_arg__min__least,axiom,
    ! [S: set_o,Y: $o,F: $o > int] :
      ( ( finite_finite_o @ S )
     => ( ( S != bot_bot_set_o )
       => ( ( member_o @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic2773365557947402859_o_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2674_arg__min__least,axiom,
    ! [S: set_nat,Y: nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( S != bot_bot_set_nat )
       => ( ( member_nat2 @ Y @ S )
         => ( ord_less_eq_int @ ( F @ ( lattic7444442490073309207at_int @ F @ S ) ) @ ( F @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2675_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( A3 != bot_bo7824918357723582233_nat_o )
         => ( ( lattic956194824204696298_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) )
            = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2676_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ~ ( member_assn @ X @ A3 )
       => ( ( A3 != bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A3 ) )
            = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2677_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ~ ( member_set_nat @ X @ A3 )
       => ( ( A3 != bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
            = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2678_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ~ ( member_Product_unit @ X @ A3 )
       => ( ( A3 != bot_bo3957492148770167129t_unit )
         => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
            = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2679_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ~ ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( A3 != bot_bo3083307316010499117at_nat )
         => ( ( lattic30941717366863870at_nat @ ( insert9200635055090092081at_nat @ X @ A3 ) )
            = ( inf_in2572325071724192079at_nat @ X @ ( lattic30941717366863870at_nat @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2680_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A3 ) )
            = ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2681_Inf__fin_Oinsert__not__elem,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat2 @ X @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A3 ) ) ) ) ) ) ).

% Inf_fin.insert_not_elem
thf(fact_2682_Inf__fin_Oclosed,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( A3 != bot_bo7824918357723582233_nat_o )
       => ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( inf_in1318976480646536635_nat_o @ X2 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X2 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
         => ( member6576561426505652726_nat_o @ ( lattic956194824204696298_nat_o @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2683_Inf__fin_Oclosed,axiom,
    ! [A3: set_assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( A3 != bot_bot_set_assn )
       => ( ! [X2: assn,Y2: assn] : ( member_assn @ ( inf_inf_assn @ X2 @ Y2 ) @ ( insert_assn @ X2 @ ( insert_assn @ Y2 @ bot_bot_set_assn ) ) )
         => ( member_assn @ ( lattic47131356835913163n_assn @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2684_Inf__fin_Oclosed,axiom,
    ! [A3: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( A3 != bot_bot_set_set_nat )
       => ( ! [X2: set_nat,Y2: set_nat] : ( member_set_nat @ ( inf_inf_set_nat @ X2 @ Y2 ) @ ( insert_set_nat @ X2 @ ( insert_set_nat @ Y2 @ bot_bot_set_set_nat ) ) )
         => ( member_set_nat @ ( lattic3014633134055518761et_nat @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2685_Inf__fin_Oclosed,axiom,
    ! [A3: set_Product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( A3 != bot_bo3957492148770167129t_unit )
       => ( ! [X2: product_unit,Y2: product_unit] : ( member_Product_unit @ ( inf_inf_Product_unit @ X2 @ Y2 ) @ ( insert_Product_unit @ X2 @ ( insert_Product_unit @ Y2 @ bot_bo3957492148770167129t_unit ) ) )
         => ( member_Product_unit @ ( lattic1263872656861969706t_unit @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2686_Inf__fin_Oclosed,axiom,
    ! [A3: set_se7855581050983116737at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( A3 != bot_bo3083307316010499117at_nat )
       => ( ! [X2: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] : ( member2643936169264416010at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y2 ) @ ( insert9200635055090092081at_nat @ X2 @ ( insert9200635055090092081at_nat @ Y2 @ bot_bo3083307316010499117at_nat ) ) )
         => ( member2643936169264416010at_nat @ ( lattic30941717366863870at_nat @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2687_Inf__fin_Oclosed,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : ( member_o @ ( inf_inf_o @ X2 @ Y2 ) @ ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic4107685809792843317_fin_o @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2688_Inf__fin_Oclosed,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( inf_inf_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
         => ( member_nat2 @ ( lattic5238388535129920115in_nat @ A3 ) @ A3 ) ) ) ) ).

% Inf_fin.closed
thf(fact_2689_mergesort__def,axiom,
    ( mergesort_assn
    = ( merges2400687674486622701l_assn @ ord_less_eq_assn ) ) ).

% mergesort_def
thf(fact_2690_mergesort__def,axiom,
    ( mergesort_nat
    = ( mergesort_by_rel_nat @ ord_less_eq_nat ) ) ).

% mergesort_def
thf(fact_2691_mergesort__def,axiom,
    ( mergesort_int
    = ( mergesort_by_rel_int @ ord_less_eq_int ) ) ).

% mergesort_def
thf(fact_2692_mergesort__def,axiom,
    ( mergesort_set_nat
    = ( merges7492048612793653835et_nat @ ord_less_eq_set_nat ) ) ).

% mergesort_def
thf(fact_2693_the__elem__eq,axiom,
    ! [X: $o] :
      ( ( the_elem_o @ ( insert_o @ X @ bot_bot_set_o ) )
      = X ) ).

% the_elem_eq
thf(fact_2694_the__elem__eq,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
      = X ) ).

% the_elem_eq
thf(fact_2695_is__singletonI,axiom,
    ! [X: $o] : ( is_singleton_o @ ( insert_o @ X @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_2696_is__singletonI,axiom,
    ! [X: nat] : ( is_singleton_nat @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_2697_subset__emptyI,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ! [X2: produc3658429121746597890et_nat > $o] :
          ~ ( member6576561426505652726_nat_o @ X2 @ A3 )
     => ( ord_le2965882846123202637_nat_o @ A3 @ bot_bo7824918357723582233_nat_o ) ) ).

% subset_emptyI
thf(fact_2698_subset__emptyI,axiom,
    ! [A3: set_o] :
      ( ! [X2: $o] :
          ~ ( member_o @ X2 @ A3 )
     => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).

% subset_emptyI
thf(fact_2699_subset__emptyI,axiom,
    ! [A3: set_nat] :
      ( ! [X2: nat] :
          ~ ( member_nat2 @ X2 @ A3 )
     => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_2700_sorted__list__of__set__nonempty,axiom,
    ! [A3: set_int] :
      ( ( finite_finite_int @ A3 )
     => ( ( A3 != bot_bot_set_int )
       => ( ( linord2612477271533052124et_int @ A3 )
          = ( cons_int @ ( lattic8718645017227715691in_int @ A3 ) @ ( linord2612477271533052124et_int @ ( minus_minus_set_int @ A3 @ ( insert_int2 @ ( lattic8718645017227715691in_int @ A3 ) @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_2701_sorted__list__of__set__nonempty,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( linord3142498349692569832_set_o @ A3 )
          = ( cons_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_2702_sorted__list__of__set__nonempty,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( linord2614967742042102400et_nat @ A3 )
          = ( cons_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ ( lattic8721135487736765967in_nat @ A3 ) @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% sorted_list_of_set_nonempty
thf(fact_2703_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
            = bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
           != bot_bot_set_assn )
         => ( ( lattic2150320897289308081n_assn @ ( insert_assn @ X @ A3 ) )
            = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_2704_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
            = bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
           != bot_bot_set_set_nat )
         => ( ( lattic3835124923745554447et_nat @ ( insert_set_nat @ X @ A3 ) )
            = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_2705_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( lattic1508158080041050831_fin_o @ ( insert_o @ X @ A3 ) )
        = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_2706_Sup__fin_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic1093996805478795353in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Sup_fin.insert_remove
thf(fact_2707_Sup__fin_Oremove,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
              = bot_bo7824918357723582233_nat_o )
           => ( ( lattic7320199455484906628_nat_o @ A3 )
              = X ) )
          & ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
             != bot_bo7824918357723582233_nat_o )
           => ( ( lattic7320199455484906628_nat_o @ A3 )
              = ( sup_su5453871518329203617_nat_o @ X @ ( lattic7320199455484906628_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_2708_Sup__fin_Oremove,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( member_assn @ X @ A3 )
       => ( ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
              = bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
             != bot_bot_set_assn )
           => ( ( lattic2150320897289308081n_assn @ A3 )
              = ( sup_sup_assn @ X @ ( lattic2150320897289308081n_assn @ ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_2709_Sup__fin_Oremove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X @ A3 )
       => ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
              = bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ A3 )
              = X ) )
          & ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
             != bot_bot_set_set_nat )
           => ( ( lattic3835124923745554447et_nat @ A3 )
              = ( sup_sup_set_nat @ X @ ( lattic3835124923745554447et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_2710_Sup__fin_Oremove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( lattic1508158080041050831_fin_o @ A3 )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( sup_sup_o @ X @ ( lattic1508158080041050831_fin_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_2711_Sup__fin_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic1093996805478795353in_nat @ A3 )
              = ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Sup_fin.remove
thf(fact_2712_Inf__fin_Oremove,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
              = bot_bo7824918357723582233_nat_o )
           => ( ( lattic956194824204696298_nat_o @ A3 )
              = X ) )
          & ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
             != bot_bo7824918357723582233_nat_o )
           => ( ( lattic956194824204696298_nat_o @ A3 )
              = ( inf_in1318976480646536635_nat_o @ X @ ( lattic956194824204696298_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2713_Inf__fin_Oremove,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( member_assn @ X @ A3 )
       => ( ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
              = bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
             != bot_bot_set_assn )
           => ( ( lattic47131356835913163n_assn @ A3 )
              = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2714_Inf__fin_Oremove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( member_set_nat @ X @ A3 )
       => ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
              = bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ A3 )
              = X ) )
          & ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
             != bot_bot_set_set_nat )
           => ( ( lattic3014633134055518761et_nat @ A3 )
              = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2715_Inf__fin_Oremove,axiom,
    ! [A3: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( member_Product_unit @ X @ A3 )
       => ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
              = bot_bo3957492148770167129t_unit )
           => ( ( lattic1263872656861969706t_unit @ A3 )
              = X ) )
          & ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
             != bot_bo3957492148770167129t_unit )
           => ( ( lattic1263872656861969706t_unit @ A3 )
              = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2716_Inf__fin_Oremove,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( member2643936169264416010at_nat @ X @ A3 )
       => ( ( ( ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
              = bot_bo3083307316010499117at_nat )
           => ( ( lattic30941717366863870at_nat @ A3 )
              = X ) )
          & ( ( ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
             != bot_bo3083307316010499117at_nat )
           => ( ( lattic30941717366863870at_nat @ A3 )
              = ( inf_in2572325071724192079at_nat @ X @ ( lattic30941717366863870at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2717_Inf__fin_Oremove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( lattic4107685809792843317_fin_o @ A3 )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2718_Inf__fin_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic5238388535129920115in_nat @ A3 )
              = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Inf_fin.remove
thf(fact_2719_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_assn,X: assn] :
      ( ( finite_finite_assn @ A3 )
     => ( ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
            = bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) )
           != bot_bot_set_assn )
         => ( ( lattic47131356835913163n_assn @ ( insert_assn @ X @ A3 ) )
            = ( inf_inf_assn @ X @ ( lattic47131356835913163n_assn @ ( minus_minus_set_assn @ A3 @ ( insert_assn @ X @ bot_bot_set_assn ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2720_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A3 )
     => ( ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
            = bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) )
           != bot_bot_set_set_nat )
         => ( ( lattic3014633134055518761et_nat @ ( insert_set_nat @ X @ A3 ) )
            = ( inf_inf_set_nat @ X @ ( lattic3014633134055518761et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2721_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_Product_unit,X: product_unit] :
      ( ( finite4290736615968046902t_unit @ A3 )
     => ( ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
            = bot_bo3957492148770167129t_unit )
         => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) )
           != bot_bo3957492148770167129t_unit )
         => ( ( lattic1263872656861969706t_unit @ ( insert_Product_unit @ X @ A3 ) )
            = ( inf_inf_Product_unit @ X @ ( lattic1263872656861969706t_unit @ ( minus_6452836326544984404t_unit @ A3 @ ( insert_Product_unit @ X @ bot_bo3957492148770167129t_unit ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2722_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_se7855581050983116737at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( finite9047747110432174090at_nat @ A3 )
     => ( ( ( ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
            = bot_bo3083307316010499117at_nat )
         => ( ( lattic30941717366863870at_nat @ ( insert9200635055090092081at_nat @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) )
           != bot_bo3083307316010499117at_nat )
         => ( ( lattic30941717366863870at_nat @ ( insert9200635055090092081at_nat @ X @ A3 ) )
            = ( inf_in2572325071724192079at_nat @ X @ ( lattic30941717366863870at_nat @ ( minus_4207664762107033000at_nat @ A3 @ ( insert9200635055090092081at_nat @ X @ bot_bo3083307316010499117at_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2723_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( lattic4107685809792843317_fin_o @ ( insert_o @ X @ A3 ) )
        = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( inf_inf_o @ X @ ( lattic4107685809792843317_fin_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2724_Inf__fin_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic5238388535129920115in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Inf_fin.insert_remove
thf(fact_2725_is__singletonE,axiom,
    ! [A3: set_o] :
      ( ( is_singleton_o @ A3 )
     => ~ ! [X2: $o] :
            ( A3
           != ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_2726_is__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( is_singleton_nat @ A3 )
     => ~ ! [X2: nat] :
            ( A3
           != ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_2727_Diff__iff,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) )
      = ( ( member6576561426505652726_nat_o @ C2 @ A3 )
        & ~ ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% Diff_iff
thf(fact_2728_DiffI,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ C2 @ B3 )
       => ( member6576561426505652726_nat_o @ C2 @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_2729_Diff__empty,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
      = A3 ) ).

% Diff_empty
thf(fact_2730_Diff__empty,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
      = A3 ) ).

% Diff_empty
thf(fact_2731_empty__Diff,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_2732_empty__Diff,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_2733_Diff__cancel,axiom,
    ! [A3: set_o] :
      ( ( minus_minus_set_o @ A3 @ A3 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_2734_Diff__cancel,axiom,
    ! [A3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ A3 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_2735_insert__Diff1,axiom,
    ! [X: $o,B3: set_o,A3: set_o] :
      ( ( member_o @ X @ B3 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A3 ) @ B3 )
        = ( minus_minus_set_o @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_2736_insert__Diff1,axiom,
    ! [X: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ B3 )
     => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) @ B3 )
        = ( minus_1801376950450012436_nat_o @ A3 @ B3 ) ) ) ).

% insert_Diff1
thf(fact_2737_Diff__insert0,axiom,
    ! [X: $o,A3: set_o,B3: set_o] :
      ( ~ ( member_o @ X @ A3 )
     => ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ B3 ) )
        = ( minus_minus_set_o @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_2738_Diff__insert0,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
     => ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ B3 ) )
        = ( minus_1801376950450012436_nat_o @ A3 @ B3 ) ) ) ).

% Diff_insert0
thf(fact_2739_Un__Diff__cancel2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B3 @ A3 ) @ A3 )
      = ( sup_sup_set_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_2740_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_2741_Diff__eq__empty__iff,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_2742_Diff__eq__empty__iff,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A3 @ B3 ) ) ).

% Diff_eq_empty_iff
thf(fact_2743_insert__Diff__single,axiom,
    ! [A: $o,A3: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_2744_insert__Diff__single,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat2 @ A @ A3 ) ) ).

% insert_Diff_single
thf(fact_2745_Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ A3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_2746_Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B3 @ A3 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_2747_Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_2748_less__assn__def,axiom,
    ( ord_less_assn
    = ( ^ [A2: assn,B2: assn] :
          ( ( ord_less_eq_assn @ A2 @ B2 )
          & ( A2 != B2 ) ) ) ) ).

% less_assn_def
thf(fact_2749_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2750_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_2751_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( A = B )
        = ( C2 = D2 ) ) ) ).

% diff_eq_diff_eq
thf(fact_2752_DiffD2,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) )
     => ~ ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ).

% DiffD2
thf(fact_2753_DiffD1,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) )
     => ( member6576561426505652726_nat_o @ C2 @ A3 ) ) ).

% DiffD1
thf(fact_2754_DiffE,axiom,
    ! [C2: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ C2 @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) )
     => ~ ( ( member6576561426505652726_nat_o @ C2 @ A3 )
         => ( member6576561426505652726_nat_o @ C2 @ B3 ) ) ) ).

% DiffE
thf(fact_2755_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_2756_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_2757_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_2758_diff__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D2 @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_mono
thf(fact_2759_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_2760_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_2761_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D2 ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C2 @ D2 ) ) ) ).

% diff_eq_diff_less
thf(fact_2762_diff__strict__mono,axiom,
    ! [A: int,B: int,D2: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D2 @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D2 ) ) ) ) ).

% diff_strict_mono
thf(fact_2763_left__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_2764_right__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_2765_left__diff__distrib_H,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C2 ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2766_left__diff__distrib_H,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_2767_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_2768_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_2769_double__diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C3 )
       => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C3 @ A3 ) )
          = A3 ) ) ) ).

% double_diff
thf(fact_2770_Diff__subset,axiom,
    ! [A3: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 ) ).

% Diff_subset
thf(fact_2771_Diff__mono,axiom,
    ! [A3: set_nat,C3: set_nat,D: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ C3 )
     => ( ( ord_less_eq_set_nat @ D @ B3 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ C3 @ D ) ) ) ) ).

% Diff_mono
thf(fact_2772_insert__Diff__if,axiom,
    ! [X: $o,B3: set_o,A3: set_o] :
      ( ( ( member_o @ X @ B3 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A3 ) @ B3 )
          = ( minus_minus_set_o @ A3 @ B3 ) ) )
      & ( ~ ( member_o @ X @ B3 )
       => ( ( minus_minus_set_o @ ( insert_o @ X @ A3 ) @ B3 )
          = ( insert_o @ X @ ( minus_minus_set_o @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2773_insert__Diff__if,axiom,
    ! [X: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ X @ B3 )
       => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) @ B3 )
          = ( minus_1801376950450012436_nat_o @ A3 @ B3 ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ X @ B3 )
       => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) @ B3 )
          = ( insert5175938949040314269_nat_o @ X @ ( minus_1801376950450012436_nat_o @ A3 @ B3 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_2774_Int__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ C3 )
      = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C3 ) ) ) ).

% Int_Diff
thf(fact_2775_Int__Diff,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ C3 )
      = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B3 @ C3 ) ) ) ).

% Int_Diff
thf(fact_2776_Diff__Int2,axiom,
    ! [A3: set_nat,C3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_2777_Diff__Int2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ B3 ) ) ).

% Diff_Int2
thf(fact_2778_Diff__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( inf_inf_set_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_2779_Diff__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ).

% Diff_Diff_Int
thf(fact_2780_Diff__Int__distrib,axiom,
    ! [C3: set_nat,A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ C3 @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C3 @ A3 ) @ ( inf_inf_set_nat @ C3 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_2781_Diff__Int__distrib,axiom,
    ! [C3: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C3 @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C3 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C3 @ B3 ) ) ) ).

% Diff_Int_distrib
thf(fact_2782_Diff__Int__distrib2,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C3 ) @ ( inf_inf_set_nat @ B3 @ C3 ) ) ) ).

% Diff_Int_distrib2
thf(fact_2783_Diff__Int__distrib2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ C3 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C3 ) @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) ) ) ).

% Diff_Int_distrib2
thf(fact_2784_set__diff__diff__left,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% set_diff_diff_left
thf(fact_2785_Un__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C3 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C3 ) @ ( minus_minus_set_nat @ B3 @ C3 ) ) ) ).

% Un_Diff
thf(fact_2786_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2453136405763929_nat_o @ A3 @ B3 )
     => ? [B5: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ B5 @ ( minus_1801376950450012436_nat_o @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_2787_diff__shunt__var,axiom,
    ! [X: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2788_diff__shunt__var,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( minus_minus_assn @ X @ Y )
        = bot_bot_assn )
      = ( ord_less_eq_assn @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2789_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2790_subset__minus__empty,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ B3 )
     => ( ( minus_minus_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% subset_minus_empty
thf(fact_2791_subset__minus__empty,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( minus_minus_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% subset_minus_empty
thf(fact_2792_Diff__insert__absorb,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
     => ( ( minus_1801376950450012436_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2793_Diff__insert__absorb,axiom,
    ! [X: $o,A3: set_o] :
      ( ~ ( member_o @ X @ A3 )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A3 ) @ ( insert_o @ X @ bot_bot_set_o ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2794_Diff__insert__absorb,axiom,
    ! [X: nat,A3: set_nat] :
      ( ~ ( member_nat2 @ X @ A3 )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A3 ) @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = A3 ) ) ).

% Diff_insert_absorb
thf(fact_2795_Diff__insert2,axiom,
    ! [A3: set_o,A: $o,B3: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B3 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_2796_Diff__insert2,axiom,
    ! [A3: set_nat,A: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) @ B3 ) ) ).

% Diff_insert2
thf(fact_2797_insert__Diff,axiom,
    ! [A: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ A @ A3 )
     => ( ( insert5175938949040314269_nat_o @ A @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ A @ bot_bo7824918357723582233_nat_o ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2798_insert__Diff,axiom,
    ! [A: $o,A3: set_o] :
      ( ( member_o @ A @ A3 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A3 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2799_insert__Diff,axiom,
    ! [A: nat,A3: set_nat] :
      ( ( member_nat2 @ A @ A3 )
     => ( ( insert_nat2 @ A @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) )
        = A3 ) ) ).

% insert_Diff
thf(fact_2800_Diff__insert,axiom,
    ! [A3: set_o,A: $o,B3: set_o] :
      ( ( minus_minus_set_o @ A3 @ ( insert_o @ A @ B3 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B3 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_2801_Diff__insert,axiom,
    ! [A3: set_nat,A: nat,B3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ A @ B3 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_2802_insert__minus__eq,axiom,
    ! [X: $o,Y: $o,A3: set_o] :
      ( ( X != Y )
     => ( ( minus_minus_set_o @ ( insert_o @ X @ A3 ) @ ( insert_o @ Y @ bot_bot_set_o ) )
        = ( insert_o @ X @ ( minus_minus_set_o @ A3 @ ( insert_o @ Y @ bot_bot_set_o ) ) ) ) ) ).

% insert_minus_eq
thf(fact_2803_insert__minus__eq,axiom,
    ! [X: nat,Y: nat,A3: set_nat] :
      ( ( X != Y )
     => ( ( minus_minus_set_nat @ ( insert_nat2 @ X @ A3 ) @ ( insert_nat2 @ Y @ bot_bot_set_nat ) )
        = ( insert_nat2 @ X @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) ) ) ) ).

% insert_minus_eq
thf(fact_2804_set__minus__singleton__eq,axiom,
    ! [X: produc3658429121746597890et_nat > $o,X5: set_Pr4532377907799695533_nat_o] :
      ( ~ ( member6576561426505652726_nat_o @ X @ X5 )
     => ( ( minus_1801376950450012436_nat_o @ X5 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_2805_set__minus__singleton__eq,axiom,
    ! [X: $o,X5: set_o] :
      ( ~ ( member_o @ X @ X5 )
     => ( ( minus_minus_set_o @ X5 @ ( insert_o @ X @ bot_bot_set_o ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_2806_set__minus__singleton__eq,axiom,
    ! [X: nat,X5: set_nat] :
      ( ~ ( member_nat2 @ X @ X5 )
     => ( ( minus_minus_set_nat @ X5 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = X5 ) ) ).

% set_minus_singleton_eq
thf(fact_2807_subset__Diff__insert,axiom,
    ! [A3: set_o,B3: set_o,X: $o,C3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B3 @ ( insert_o @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B3 @ C3 ) )
        & ~ ( member_o @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2808_subset__Diff__insert,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,B3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,C3: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ ( minus_1801376950450012436_nat_o @ B3 @ ( insert5175938949040314269_nat_o @ X @ C3 ) ) )
      = ( ( ord_le2965882846123202637_nat_o @ A3 @ ( minus_1801376950450012436_nat_o @ B3 @ C3 ) )
        & ~ ( member6576561426505652726_nat_o @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2809_subset__Diff__insert,axiom,
    ! [A3: set_nat,B3: set_nat,X: nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ ( insert_nat2 @ X @ C3 ) ) )
      = ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ C3 ) )
        & ~ ( member_nat2 @ X @ A3 ) ) ) ).

% subset_Diff_insert
thf(fact_2810_Int__Diff__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_2811_Int__Diff__disjoint,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B3 ) @ ( minus_minus_set_o @ A3 @ B3 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_2812_Int__Diff__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_2813_Diff__triv,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_2814_Diff__triv,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_2815_Diff__triv,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 ) ) ).

% Diff_triv
thf(fact_2816_disjoint__alt__simp1,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
        = bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2817_disjoint__alt__simp1,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
        = bot_bot_set_o ) ) ).

% disjoint_alt_simp1
thf(fact_2818_disjoint__alt__simp1,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
        = A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
        = bot_bot_set_nat ) ) ).

% disjoint_alt_simp1
thf(fact_2819_disjoint__alt__simp2,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2820_disjoint__alt__simp2,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ( minus_minus_set_o @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp2
thf(fact_2821_disjoint__alt__simp2,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ( minus_minus_set_nat @ A3 @ B3 )
       != A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp2
thf(fact_2822_Diff__subset__conv,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ C3 )
      = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) ) ) ).

% Diff_subset_conv
thf(fact_2823_Diff__partition,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ B3 )
     => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
        = B3 ) ) ).

% Diff_partition
thf(fact_2824_Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B3 @ C3 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_2825_Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C3 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C3 ) ) ) ).

% Diff_Un
thf(fact_2826_Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C3: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B3 @ C3 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_2827_Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat,C3: set_nat] :
      ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B3 @ C3 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ C3 ) ) ) ).

% Diff_Int
thf(fact_2828_Int__Diff__Un,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_2829_Int__Diff__Un,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B3 ) @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Int_Diff_Un
thf(fact_2830_Un__Diff__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_2831_Un__Diff__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ ( inf_inf_set_nat @ A3 @ B3 ) )
      = A3 ) ).

% Un_Diff_Int
thf(fact_2832_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2833_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat2 @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_2834_is__singletonI_H,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o] :
      ( ( A3 != bot_bo7824918357723582233_nat_o )
     => ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] :
            ( ( member6576561426505652726_nat_o @ X2 @ A3 )
           => ( ( member6576561426505652726_nat_o @ Y2 @ A3 )
             => ( X2 = Y2 ) ) )
       => ( is_sin5180296473474724033_nat_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2835_is__singletonI_H,axiom,
    ! [A3: set_o] :
      ( ( A3 != bot_bot_set_o )
     => ( ! [X2: $o,Y2: $o] :
            ( ( member_o @ X2 @ A3 )
           => ( ( member_o @ Y2 @ A3 )
             => ( X2 = Y2 ) ) )
       => ( is_singleton_o @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2836_is__singletonI_H,axiom,
    ! [A3: set_nat] :
      ( ( A3 != bot_bot_set_nat )
     => ( ! [X2: nat,Y2: nat] :
            ( ( member_nat2 @ X2 @ A3 )
           => ( ( member_nat2 @ Y2 @ A3 )
             => ( X2 = Y2 ) ) )
       => ( is_singleton_nat @ A3 ) ) ) ).

% is_singletonI'
thf(fact_2837_infinite__remove,axiom,
    ! [S: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_2838_infinite__remove,axiom,
    ! [S: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_2839_infinite__coinduct,axiom,
    ! [X5: set_o > $o,A3: set_o] :
      ( ( X5 @ A3 )
     => ( ! [A7: set_o] :
            ( ( X5 @ A7 )
           => ? [X6: $o] :
                ( ( member_o @ X6 @ A7 )
                & ( ( X5 @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2840_infinite__coinduct,axiom,
    ! [X5: set_nat > $o,A3: set_nat] :
      ( ( X5 @ A3 )
     => ( ! [A7: set_nat] :
            ( ( X5 @ A7 )
           => ? [X6: nat] :
                ( ( member_nat2 @ X6 @ A7 )
                & ( ( X5 @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A3 ) ) ) ).

% infinite_coinduct
thf(fact_2841_finite__empty__induct,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: produc3658429121746597890et_nat > $o,A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( member6576561426505652726_nat_o @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ A4 @ bot_bo7824918357723582233_nat_o ) ) ) ) ) )
         => ( P @ bot_bo7824918357723582233_nat_o ) ) ) ) ).

% finite_empty_induct
thf(fact_2842_finite__empty__induct,axiom,
    ! [A3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: $o,A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( member_o @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_2843_finite__empty__induct,axiom,
    ! [A3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A3 )
     => ( ( P @ A3 )
       => ( ! [A4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat2 @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ A4 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_2844_Diff__single__insert,axiom,
    ! [A3: set_o,X: $o,B3: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B3 )
     => ( ord_less_eq_set_o @ A3 @ ( insert_o @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_2845_Diff__single__insert,axiom,
    ! [A3: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B3 )
     => ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X @ B3 ) ) ) ).

% Diff_single_insert
thf(fact_2846_subset__insert__iff,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2965882846123202637_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ B3 ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ A3 )
         => ( ord_le2965882846123202637_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B3 ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
         => ( ord_le2965882846123202637_nat_o @ A3 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_2847_subset__insert__iff,axiom,
    ! [A3: set_o,X: $o,B3: set_o] :
      ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X @ B3 ) )
      = ( ( ( member_o @ X @ A3 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B3 ) )
        & ( ~ ( member_o @ X @ A3 )
         => ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_2848_subset__insert__iff,axiom,
    ! [A3: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat2 @ X @ B3 ) )
      = ( ( ( member_nat2 @ X @ A3 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B3 ) )
        & ( ~ ( member_nat2 @ X @ A3 )
         => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ).

% subset_insert_iff
thf(fact_2849_remove__subset,axiom,
    ! [X: produc3658429121746597890et_nat > $o,S: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ S )
     => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ S @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ S ) ) ).

% remove_subset
thf(fact_2850_remove__subset,axiom,
    ! [X: $o,S: set_o] :
      ( ( member_o @ X @ S )
     => ( ord_less_set_o @ ( minus_minus_set_o @ S @ ( insert_o @ X @ bot_bot_set_o ) ) @ S ) ) ).

% remove_subset
thf(fact_2851_remove__subset,axiom,
    ! [X: nat,S: set_nat] :
      ( ( member_nat2 @ X @ S )
     => ( ord_less_set_nat @ ( minus_minus_set_nat @ S @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ S ) ) ).

% remove_subset
thf(fact_2852_disjoint__alt__simp3,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
       != bot_bo2099793752762293965at_nat ) ) ).

% disjoint_alt_simp3
thf(fact_2853_disjoint__alt__simp3,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_o @ A3 @ B3 )
       != bot_bot_set_o ) ) ).

% disjoint_alt_simp3
thf(fact_2854_disjoint__alt__simp3,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ B3 ) @ A3 )
      = ( ( inf_inf_set_nat @ A3 @ B3 )
       != bot_bot_set_nat ) ) ).

% disjoint_alt_simp3
thf(fact_2855_finite__remove__induct,axiom,
    ! [B3: set_Pr4532377907799695533_nat_o,P: set_Pr4532377907799695533_nat_o > $o] :
      ( ( finite3252695134891459830_nat_o @ B3 )
     => ( ( P @ bot_bo7824918357723582233_nat_o )
       => ( ! [A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( A7 != bot_bo7824918357723582233_nat_o )
               => ( ( ord_le2965882846123202637_nat_o @ A7 @ B3 )
                 => ( ! [X6: produc3658429121746597890et_nat > $o] :
                        ( ( member6576561426505652726_nat_o @ X6 @ A7 )
                       => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_2856_finite__remove__induct,axiom,
    ! [B3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B3 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_2857_finite__remove__induct,axiom,
    ! [B3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B3 )
                 => ( ! [X6: nat] :
                        ( ( member_nat2 @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% finite_remove_induct
thf(fact_2858_remove__induct,axiom,
    ! [P: set_Pr4532377907799695533_nat_o > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( P @ bot_bo7824918357723582233_nat_o )
     => ( ( ~ ( finite3252695134891459830_nat_o @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_Pr4532377907799695533_nat_o] :
              ( ( finite3252695134891459830_nat_o @ A7 )
             => ( ( A7 != bot_bo7824918357723582233_nat_o )
               => ( ( ord_le2965882846123202637_nat_o @ A7 @ B3 )
                 => ( ! [X6: produc3658429121746597890et_nat > $o] :
                        ( ( member6576561426505652726_nat_o @ X6 @ A7 )
                       => ( P @ ( minus_1801376950450012436_nat_o @ A7 @ ( insert5175938949040314269_nat_o @ X6 @ bot_bo7824918357723582233_nat_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_2859_remove__induct,axiom,
    ! [P: set_o > $o,B3: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_o] :
              ( ( finite_finite_o @ A7 )
             => ( ( A7 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A7 @ B3 )
                 => ( ! [X6: $o] :
                        ( ( member_o @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_o @ A7 @ ( insert_o @ X6 @ bot_bot_set_o ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_2860_remove__induct,axiom,
    ! [P: set_nat > $o,B3: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B3 )
         => ( P @ B3 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B3 )
                 => ( ! [X6: nat] :
                        ( ( member_nat2 @ X6 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat2 @ X6 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B3 ) ) ) ) ).

% remove_induct
thf(fact_2861_finite__induct__select,axiom,
    ! [S: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T3: set_o] :
              ( ( ord_less_set_o @ T3 @ S )
             => ( ( P @ T3 )
               => ? [X6: $o] :
                    ( ( member_o @ X6 @ ( minus_minus_set_o @ S @ T3 ) )
                    & ( P @ ( insert_o @ X6 @ T3 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2862_finite__induct__select,axiom,
    ! [S: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T3: set_nat] :
              ( ( ord_less_set_nat @ T3 @ S )
             => ( ( P @ T3 )
               => ? [X6: nat] :
                    ( ( member_nat2 @ X6 @ ( minus_minus_set_nat @ S @ T3 ) )
                    & ( P @ ( insert_nat2 @ X6 @ T3 ) ) ) ) )
         => ( P @ S ) ) ) ) ).

% finite_induct_select
thf(fact_2863_psubset__insert__iff,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,B3: set_Pr4532377907799695533_nat_o] :
      ( ( ord_le2453136405763929_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ B3 ) )
      = ( ( ( member6576561426505652726_nat_o @ X @ B3 )
         => ( ord_le2453136405763929_nat_o @ A3 @ B3 ) )
        & ( ~ ( member6576561426505652726_nat_o @ X @ B3 )
         => ( ( ( member6576561426505652726_nat_o @ X @ A3 )
             => ( ord_le2453136405763929_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ B3 ) )
            & ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
             => ( ord_le2965882846123202637_nat_o @ A3 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2864_psubset__insert__iff,axiom,
    ! [A3: set_o,X: $o,B3: set_o] :
      ( ( ord_less_set_o @ A3 @ ( insert_o @ X @ B3 ) )
      = ( ( ( member_o @ X @ B3 )
         => ( ord_less_set_o @ A3 @ B3 ) )
        & ( ~ ( member_o @ X @ B3 )
         => ( ( ( member_o @ X @ A3 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) @ B3 ) )
            & ( ~ ( member_o @ X @ A3 )
             => ( ord_less_eq_set_o @ A3 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2865_psubset__insert__iff,axiom,
    ! [A3: set_nat,X: nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ ( insert_nat2 @ X @ B3 ) )
      = ( ( ( member_nat2 @ X @ B3 )
         => ( ord_less_set_nat @ A3 @ B3 ) )
        & ( ~ ( member_nat2 @ X @ B3 )
         => ( ( ( member_nat2 @ X @ A3 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ B3 ) )
            & ( ~ ( member_nat2 @ X @ A3 )
             => ( ord_less_eq_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_2866_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X3: $o] :
          ( A5
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_2867_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X3: nat] :
          ( A5
          = ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_2868_inf__period_I2_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X2: int,K2: int] :
          ( ( P @ X2 )
          = ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X2: int,K2: int] :
            ( ( Q @ X2 )
            = ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X6: int,K3: int] :
            ( ( ( P @ X6 )
              | ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) )
              | ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2869_inf__period_I1_J,axiom,
    ! [P: int > $o,D: int,Q: int > $o] :
      ( ! [X2: int,K2: int] :
          ( ( P @ X2 )
          = ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D ) ) ) )
     => ( ! [X2: int,K2: int] :
            ( ( Q @ X2 )
            = ( Q @ ( minus_minus_int @ X2 @ ( times_times_int @ K2 @ D ) ) ) )
       => ! [X6: int,K3: int] :
            ( ( ( P @ X6 )
              & ( Q @ X6 ) )
            = ( ( P @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) )
              & ( Q @ ( minus_minus_int @ X6 @ ( times_times_int @ K3 @ D ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2870_remove__def,axiom,
    ( remove_o
    = ( ^ [X3: $o,A5: set_o] : ( minus_minus_set_o @ A5 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% remove_def
thf(fact_2871_remove__def,axiom,
    ( remove_nat
    = ( ^ [X3: nat,A5: set_nat] : ( minus_minus_set_nat @ A5 @ ( insert_nat2 @ X3 @ bot_bot_set_nat ) ) ) ) ).

% remove_def
thf(fact_2872_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( linord3142498349692569832_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( remove1_o @ X @ ( linord3142498349692569832_set_o @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_2873_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( linord2614967742042102400et_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        = ( remove1_nat @ X @ ( linord2614967742042102400et_nat @ A3 ) ) ) ) ).

% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_2874_Min_Oremove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( lattic1973801136483472281_Min_o @ A3 )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Min.remove
thf(fact_2875_Min_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic8721135487736765967in_nat @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic8721135487736765967in_nat @ A3 )
              = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Min.remove
thf(fact_2876_Min_Oinsert__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ A3 ) )
        = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Min.insert_remove
thf(fact_2877_Min_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Min.insert_remove
thf(fact_2878_Max_Oremove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( lattic1921953407002678535_Max_o @ A3 )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_2879_Max_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A3 )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ A3 )
              = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% Max.remove
thf(fact_2880_Max_Oinsert__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A3 ) )
        = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
              = bot_bot_set_o )
           => X )
          & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
             != bot_bot_set_o )
           => ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_2881_Max_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
            = bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A3 ) )
            = X ) )
        & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
           != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ).

% Max.insert_remove
thf(fact_2882_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5111740090977526247t_unit @ inf_inf_Product_unit @ ord_le3221252021190050221t_unit @ ord_le361264281704409273t_unit ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2883_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6529551498584149819at_nat @ inf_in2572325071724192079at_nat @ ord_le3146513528884898305at_nat @ ord_le7866589430770878221at_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2884_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic5623431474481994958t_assn @ inf_inf_assn @ ord_less_eq_assn @ ord_less_assn ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2885_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6009151579333465974et_nat @ inf_inf_nat @ ord_less_eq_nat @ ord_less_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2886_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic6006661108824415698et_int @ inf_inf_int @ ord_less_eq_int @ ord_less_int ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2887_Inf__fin_Osemilattice__order__set__axioms,axiom,
    lattic3109210760196336428et_nat @ inf_inf_set_nat @ ord_less_eq_set_nat @ ord_less_set_nat ).

% Inf_fin.semilattice_order_set_axioms
thf(fact_2888_Min__insert,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ A3 ) )
          = ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) ) ) ) ) ).

% Min_insert
thf(fact_2889_Min__insert,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A3 ) )
          = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).

% Min_insert
thf(fact_2890_min_Oidem,axiom,
    ! [A: nat] :
      ( ( ord_min_nat @ A @ A )
      = A ) ).

% min.idem
thf(fact_2891_min_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_min_nat @ A @ ( ord_min_nat @ A @ B ) )
      = ( ord_min_nat @ A @ B ) ) ).

% min.left_idem
thf(fact_2892_min_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ B )
      = ( ord_min_nat @ A @ B ) ) ).

% min.right_idem
thf(fact_2893_member__remove,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ ( remove4651630035290841522_nat_o @ Y @ A3 ) )
      = ( ( member6576561426505652726_nat_o @ X @ A3 )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_2894_min_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_2895_min_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb1
thf(fact_2896_min_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_2897_min_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb2
thf(fact_2898_min_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_2899_min_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C2 ) ) ) ).

% min.bounded_iff
thf(fact_2900_min__arg__le_I2_J,axiom,
    ! [M2: assn,N: assn] :
      ( ( ord_less_eq_assn @ M2 @ ( ord_min_assn @ M2 @ N ) )
      = ( ( ord_min_assn @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2901_min__arg__le_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( ord_min_nat @ M2 @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2902_min__arg__le_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( ord_min_int @ M2 @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2903_min__arg__le_I2_J,axiom,
    ! [M2: set_nat,N: set_nat] :
      ( ( ord_less_eq_set_nat @ M2 @ ( ord_min_set_nat @ M2 @ N ) )
      = ( ( ord_min_set_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_le(2)
thf(fact_2904_min__arg__le_I1_J,axiom,
    ! [N: assn,M2: assn] :
      ( ( ord_less_eq_assn @ N @ ( ord_min_assn @ M2 @ N ) )
      = ( ( ord_min_assn @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2905_min__arg__le_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ ( ord_min_nat @ M2 @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2906_min__arg__le_I1_J,axiom,
    ! [N: int,M2: int] :
      ( ( ord_less_eq_int @ N @ ( ord_min_int @ M2 @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2907_min__arg__le_I1_J,axiom,
    ! [N: set_nat,M2: set_nat] :
      ( ( ord_less_eq_set_nat @ N @ ( ord_min_set_nat @ M2 @ N ) )
      = ( ( ord_min_set_nat @ M2 @ N )
        = N ) ) ).

% min_arg_le(1)
thf(fact_2908_min__eq__arg_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_min_nat @ M2 @ N )
        = N )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_2909_min__eq__arg_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ( ord_min_int @ M2 @ N )
        = N )
      = ( ord_less_eq_int @ N @ M2 ) ) ).

% min_eq_arg(2)
thf(fact_2910_min__eq__arg_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_min_nat @ M2 @ N )
        = M2 )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% min_eq_arg(1)
thf(fact_2911_min__eq__arg_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( ( ord_min_int @ M2 @ N )
        = M2 )
      = ( ord_less_eq_int @ M2 @ N ) ) ).

% min_eq_arg(1)
thf(fact_2912_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2913_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2914_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2915_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2916_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_2917_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_2918_min_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_2919_min_Oabsorb3,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min.absorb3
thf(fact_2920_min_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_2921_min_Oabsorb4,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min.absorb4
thf(fact_2922_min__less__iff__conj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_min_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        & ( ord_less_nat @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2923_min__less__iff__conj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_min_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        & ( ord_less_int @ Z @ Y ) ) ) ).

% min_less_iff_conj
thf(fact_2924_min__simps_I2_J,axiom,
    ! [B: assn,A: assn] :
      ( ( ord_less_assn @ B @ A )
     => ( ( ord_min_assn @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_2925_min__simps_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_min_nat @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_2926_min__simps_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_min_int @ A @ B )
        = B ) ) ).

% min_simps(2)
thf(fact_2927_min__simps_I1_J,axiom,
    ! [A: assn,B: assn] :
      ( ( ord_less_assn @ A @ B )
     => ( ( ord_min_assn @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_2928_min__simps_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_2929_min__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_min_int @ A @ B )
        = A ) ) ).

% min_simps(1)
thf(fact_2930_min__less__self__conv_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_2931_min__less__self__conv_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( ord_min_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ B ) ) ).

% min_less_self_conv(2)
thf(fact_2932_min__less__self__conv_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ A )
      = ( ord_less_nat @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_2933_min__less__self__conv_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( ord_min_int @ A @ B ) @ A )
      = ( ord_less_int @ B @ A ) ) ).

% min_less_self_conv(1)
thf(fact_2934_min__arg__not__ge_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N ) @ N ) )
      = ( ( ord_min_nat @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2935_min__arg__not__ge_I2_J,axiom,
    ! [M2: int,N: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N ) @ N ) )
      = ( ( ord_min_int @ M2 @ N )
        = N ) ) ).

% min_arg_not_ge(2)
thf(fact_2936_min__arg__not__ge_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ ( ord_min_nat @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_nat @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2937_min__arg__not__ge_I1_J,axiom,
    ! [M2: int,N: int] :
      ( ( ~ ( ord_less_int @ ( ord_min_int @ M2 @ N ) @ M2 ) )
      = ( ( ord_min_int @ M2 @ N )
        = M2 ) ) ).

% min_arg_not_ge(1)
thf(fact_2938_max_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_2939_max_Oabsorb3,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb3
thf(fact_2940_max_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_2941_max_Oabsorb4,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb4
thf(fact_2942_max__less__iff__conj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        & ( ord_less_nat @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_2943_max__less__iff__conj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        & ( ord_less_int @ Y @ Z ) ) ) ).

% max_less_iff_conj
thf(fact_2944_min__bot2,axiom,
    ! [X: assn] :
      ( ( ord_min_assn @ X @ bot_bot_assn )
      = bot_bot_assn ) ).

% min_bot2
thf(fact_2945_min__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ X @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% min_bot2
thf(fact_2946_min__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ X @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% min_bot2
thf(fact_2947_min__bot2,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ X @ bot_bot_nat )
      = bot_bot_nat ) ).

% min_bot2
thf(fact_2948_min__bot,axiom,
    ! [X: assn] :
      ( ( ord_min_assn @ bot_bot_assn @ X )
      = bot_bot_assn ) ).

% min_bot
thf(fact_2949_min__bot,axiom,
    ! [X: set_o] :
      ( ( ord_min_set_o @ bot_bot_set_o @ X )
      = bot_bot_set_o ) ).

% min_bot
thf(fact_2950_min__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_min_set_nat @ bot_bot_set_nat @ X )
      = bot_bot_set_nat ) ).

% min_bot
thf(fact_2951_min__bot,axiom,
    ! [X: nat] :
      ( ( ord_min_nat @ bot_bot_nat @ X )
      = bot_bot_nat ) ).

% min_bot
thf(fact_2952_max__bot2,axiom,
    ! [X: assn] :
      ( ( ord_max_assn @ X @ bot_bot_assn )
      = X ) ).

% max_bot2
thf(fact_2953_max__bot2,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ X @ bot_bot_set_o )
      = X ) ).

% max_bot2
thf(fact_2954_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_2955_max__bot,axiom,
    ! [X: assn] :
      ( ( ord_max_assn @ bot_bot_assn @ X )
      = X ) ).

% max_bot
thf(fact_2956_max__bot,axiom,
    ! [X: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X )
      = X ) ).

% max_bot
thf(fact_2957_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_2958_max__min__same_I4_J,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_max_nat @ Y @ ( ord_min_nat @ X @ Y ) )
      = Y ) ).

% max_min_same(4)
thf(fact_2959_max__min__same_I3_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ Y )
      = Y ) ).

% max_min_same(3)
thf(fact_2960_max__min__same_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ X @ Y ) @ X )
      = X ) ).

% max_min_same(2)
thf(fact_2961_max__min__same_I1_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_max_nat @ X @ ( ord_min_nat @ X @ Y ) )
      = X ) ).

% max_min_same(1)
thf(fact_2962_Max__insert,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A3 ) )
          = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_2963_Max__insert,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A3 ) )
          = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ).

% Max_insert
thf(fact_2964_min_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_min_nat @ ( ord_min_nat @ A @ B ) @ C2 )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ).

% min.assoc
thf(fact_2965_min_Ocommute,axiom,
    ( ord_min_nat
    = ( ^ [A2: nat,B2: nat] : ( ord_min_nat @ B2 @ A2 ) ) ) ).

% min.commute
thf(fact_2966_max__min__distrib1,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_max_nat @ ( ord_min_nat @ B @ C2 ) @ A )
      = ( ord_min_nat @ ( ord_max_nat @ B @ A ) @ ( ord_max_nat @ C2 @ A ) ) ) ).

% max_min_distrib1
thf(fact_2967_max__min__distrib2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_max_nat @ A @ ( ord_min_nat @ B @ C2 ) )
      = ( ord_min_nat @ ( ord_max_nat @ A @ B ) @ ( ord_max_nat @ A @ C2 ) ) ) ).

% max_min_distrib2
thf(fact_2968_min_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_min_nat @ B @ ( ord_min_nat @ A @ C2 ) )
      = ( ord_min_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ).

% min.left_commute
thf(fact_2969_min__max__distrib1,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_min_nat @ ( ord_max_nat @ B @ C2 ) @ A )
      = ( ord_max_nat @ ( ord_min_nat @ B @ A ) @ ( ord_min_nat @ C2 @ A ) ) ) ).

% min_max_distrib1
thf(fact_2970_min__max__distrib2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_min_nat @ A @ ( ord_max_nat @ B @ C2 ) )
      = ( ord_max_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ A @ C2 ) ) ) ).

% min_max_distrib2
thf(fact_2971_max_Omono,axiom,
    ! [C2: nat,A: nat,D2: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D2 @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D2 ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2972_max_Omono,axiom,
    ! [C2: int,A: int,D2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D2 @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C2 @ D2 ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2973_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_2974_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_2975_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_2976_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_2977_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_2978_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_2979_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_2980_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_2981_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( A2
          = ( ord_max_nat @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_2982_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( A2
          = ( ord_max_int @ A2 @ B2 ) ) ) ) ).

% max.order_iff
thf(fact_2983_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_2984_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_2985_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_2986_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_2987_le__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z @ X )
        | ( ord_less_eq_nat @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_2988_le__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z @ X )
        | ( ord_less_eq_int @ Z @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_2989_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_2990_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = A2 ) ) ) ).

% max.absorb_iff1
thf(fact_2991_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_max_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_2992_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_max_int @ A2 @ B2 )
          = B2 ) ) ) ).

% max.absorb_iff2
thf(fact_2993_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_2994_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_2995_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_2996_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_2997_max__def,axiom,
    ( ord_max_assn
    = ( ^ [A2: assn,B2: assn] : ( if_assn @ ( ord_less_eq_assn @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_2998_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_2999_max__def,axiom,
    ( ord_max_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_3000_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ B2 @ A2 ) ) ) ).

% max_def
thf(fact_3001_max__absorb1,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_max_assn @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_3002_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_3003_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_3004_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_3005_max__absorb2,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_max_assn @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3006_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3007_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3008_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3009_less__max__iff__disj,axiom,
    ! [Z: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ Z @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Z @ X )
        | ( ord_less_nat @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3010_less__max__iff__disj,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ Z @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_int @ Z @ X )
        | ( ord_less_int @ Z @ Y ) ) ) ).

% less_max_iff_disj
thf(fact_3011_max_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3012_max_Ostrict__boundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_int @ B @ A )
         => ~ ( ord_less_int @ C2 @ A ) ) ) ).

% max.strict_boundedE
thf(fact_3013_max_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( A2
            = ( ord_max_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3014_max_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A2: int] :
          ( ( A2
            = ( ord_max_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
thf(fact_3015_max_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3016_max_Ostrict__coboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ A )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI1
thf(fact_3017_max_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3018_max_Ostrict__coboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_int @ C2 @ B )
     => ( ord_less_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.strict_coboundedI2
thf(fact_3019_min_Omono,axiom,
    ! [A: nat,C2: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ ( ord_min_nat @ C2 @ D2 ) ) ) ) ).

% min.mono
thf(fact_3020_min_Omono,axiom,
    ! [A: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D2 )
       => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ ( ord_min_int @ C2 @ D2 ) ) ) ) ).

% min.mono
thf(fact_3021_min_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( ord_min_nat @ A @ B ) ) ) ).

% min.orderE
thf(fact_3022_min_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( ord_min_int @ A @ B ) ) ) ).

% min.orderE
thf(fact_3023_min_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_min_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% min.orderI
thf(fact_3024_min_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_min_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% min.orderI
thf(fact_3025_min_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% min.boundedE
thf(fact_3026_min_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).

% min.boundedE
thf(fact_3027_min_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C2 )
       => ( ord_less_eq_nat @ A @ ( ord_min_nat @ B @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_3028_min_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C2 )
       => ( ord_less_eq_int @ A @ ( ord_min_int @ B @ C2 ) ) ) ) ).

% min.boundedI
thf(fact_3029_min_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( A2
          = ( ord_min_nat @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_3030_min_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( A2
          = ( ord_min_int @ A2 @ B2 ) ) ) ) ).

% min.order_iff
thf(fact_3031_min_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_3032_min_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ A ) ).

% min.cobounded1
thf(fact_3033_min_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_3034_min_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ B ) ).

% min.cobounded2
thf(fact_3035_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( ord_min_nat @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_3036_min_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] :
          ( ( ord_min_int @ A2 @ B2 )
          = A2 ) ) ) ).

% min.absorb_iff1
thf(fact_3037_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A2: nat] :
          ( ( ord_min_nat @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_3038_min_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A2: int] :
          ( ( ord_min_int @ A2 @ B2 )
          = B2 ) ) ) ).

% min.absorb_iff2
thf(fact_3039_min_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_3040_min_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).

% min.coboundedI1
thf(fact_3041_min_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C2 )
     => ( ord_less_eq_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_3042_min_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ B @ C2 )
     => ( ord_less_eq_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).

% min.coboundedI2
thf(fact_3043_min__le__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_eq_nat @ X @ Z )
        | ( ord_less_eq_nat @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_3044_min__le__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_eq_int @ X @ Z )
        | ( ord_less_eq_int @ Y @ Z ) ) ) ).

% min_le_iff_disj
thf(fact_3045_min__def,axiom,
    ( ord_min_assn
    = ( ^ [A2: assn,B2: assn] : ( if_assn @ ( ord_less_eq_assn @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_3046_min__def,axiom,
    ( ord_min_nat
    = ( ^ [A2: nat,B2: nat] : ( if_nat @ ( ord_less_eq_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_3047_min__def,axiom,
    ( ord_min_int
    = ( ^ [A2: int,B2: int] : ( if_int @ ( ord_less_eq_int @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_3048_min__def,axiom,
    ( ord_min_set_nat
    = ( ^ [A2: set_nat,B2: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A2 @ B2 ) @ A2 @ B2 ) ) ) ).

% min_def
thf(fact_3049_min__absorb1,axiom,
    ! [X: assn,Y: assn] :
      ( ( ord_less_eq_assn @ X @ Y )
     => ( ( ord_min_assn @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_3050_min__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_min_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_3051_min__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_min_int @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_3052_min__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_min_set_nat @ X @ Y )
        = X ) ) ).

% min_absorb1
thf(fact_3053_min__absorb2,axiom,
    ! [Y: assn,X: assn] :
      ( ( ord_less_eq_assn @ Y @ X )
     => ( ( ord_min_assn @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_3054_min__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_min_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_3055_min__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_min_int @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_3056_min__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_min_set_nat @ X @ Y )
        = Y ) ) ).

% min_absorb2
thf(fact_3057_min__less__iff__disj,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ( ord_less_nat @ X @ Z )
        | ( ord_less_nat @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_3058_min__less__iff__disj,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( ord_less_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ( ord_less_int @ X @ Z )
        | ( ord_less_int @ Y @ Z ) ) ) ).

% min_less_iff_disj
thf(fact_3059_min_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( ord_min_nat @ B @ C2 ) )
     => ~ ( ( ord_less_nat @ A @ B )
         => ~ ( ord_less_nat @ A @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_3060_min_Ostrict__boundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ ( ord_min_int @ B @ C2 ) )
     => ~ ( ( ord_less_int @ A @ B )
         => ~ ( ord_less_int @ A @ C2 ) ) ) ).

% min.strict_boundedE
thf(fact_3061_min_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat,B2: nat] :
          ( ( A2
            = ( ord_min_nat @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_3062_min_Ostrict__order__iff,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] :
          ( ( A2
            = ( ord_min_int @ A2 @ B2 ) )
          & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
thf(fact_3063_min_Ostrict__coboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ A @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_3064_min_Ostrict__coboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ C2 )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).

% min.strict_coboundedI1
thf(fact_3065_min_Ostrict__coboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ B @ C2 )
     => ( ord_less_nat @ ( ord_min_nat @ A @ B ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_3066_min_Ostrict__coboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_int @ B @ C2 )
     => ( ord_less_int @ ( ord_min_int @ A @ B ) @ C2 ) ) ).

% min.strict_coboundedI2
thf(fact_3067_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% max_diff_distrib_left
thf(fact_3068_min__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( minus_minus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( minus_minus_int @ X @ Z ) @ ( minus_minus_int @ Y @ Z ) ) ) ).

% min_diff_distrib_left
thf(fact_3069_inf__min,axiom,
    inf_inf_Product_unit = ord_min_Product_unit ).

% inf_min
thf(fact_3070_inf__min,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_min
thf(fact_3071_remove1_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1_nat @ X @ ( cons_nat @ Y @ Xs ) )
          = ( cons_nat @ Y @ ( remove1_nat @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_3072_remove1_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( remove1_int @ X @ ( cons_int @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1_int @ X @ ( cons_int @ Y @ Xs ) )
          = ( cons_int @ Y @ ( remove1_int @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_3073_remove1_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( ( remove1670527618125605709n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = Xs ) )
      & ( ( X != Y )
       => ( ( remove1670527618125605709n_assn @ X @ ( cons_P2971678138204555879n_assn @ Y @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ Y @ ( remove1670527618125605709n_assn @ X @ Xs ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_3074_remove1_Osimps_I1_J,axiom,
    ! [X: a] :
      ( ( remove1_a @ X @ nil_a )
      = nil_a ) ).

% remove1.simps(1)
thf(fact_3075_remove1_Osimps_I1_J,axiom,
    ! [X: b] :
      ( ( remove1_b @ X @ nil_b )
      = nil_b ) ).

% remove1.simps(1)
thf(fact_3076_remove1_Osimps_I1_J,axiom,
    ! [X: produc6575502325842934193n_assn] :
      ( ( remove1670527618125605709n_assn @ X @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% remove1.simps(1)
thf(fact_3077_remove1_Osimps_I1_J,axiom,
    ! [X: nat] :
      ( ( remove1_nat @ X @ nil_nat )
      = nil_nat ) ).

% remove1.simps(1)
thf(fact_3078_remove1_Osimps_I1_J,axiom,
    ! [X: int] :
      ( ( remove1_int @ X @ nil_int )
      = nil_int ) ).

% remove1.simps(1)
thf(fact_3079_Max_Osubset,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( B3 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B3 @ A3 )
         => ( ( ord_max_o @ ( lattic1921953407002678535_Max_o @ B3 ) @ ( lattic1921953407002678535_Max_o @ A3 ) )
            = ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_3080_Max_Osubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A3 )
         => ( ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ B3 ) @ ( lattic8265883725875713057ax_nat @ A3 ) )
            = ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ).

% Max.subset
thf(fact_3081_Max_Oclosed,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : ( member_o @ ( ord_max_o @ X2 @ Y2 ) @ ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_3082_Max_Oclosed,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( ord_max_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
         => ( member_nat2 @ ( lattic8265883725875713057ax_nat @ A3 ) @ A3 ) ) ) ) ).

% Max.closed
thf(fact_3083_Max_Oinsert__not__elem,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic1921953407002678535_Max_o @ ( insert_o @ X @ A3 ) )
            = ( ord_max_o @ X @ ( lattic1921953407002678535_Max_o @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_3084_Max_Oinsert__not__elem,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat2 @ X @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic8265883725875713057ax_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( ord_max_nat @ X @ ( lattic8265883725875713057ax_nat @ A3 ) ) ) ) ) ) ).

% Max.insert_not_elem
thf(fact_3085_Max_Ounion,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ( B3 != bot_bot_set_o )
           => ( ( lattic1921953407002678535_Max_o @ ( sup_sup_set_o @ A3 @ B3 ) )
              = ( ord_max_o @ ( lattic1921953407002678535_Max_o @ A3 ) @ ( lattic1921953407002678535_Max_o @ B3 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_3086_Max_Ounion,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic8265883725875713057ax_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
              = ( ord_max_nat @ ( lattic8265883725875713057ax_nat @ A3 ) @ ( lattic8265883725875713057ax_nat @ B3 ) ) ) ) ) ) ) ).

% Max.union
thf(fact_3087_Min_Osubset,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( B3 != bot_bot_set_o )
       => ( ( ord_less_eq_set_o @ B3 @ A3 )
         => ( ( ord_min_o @ ( lattic1973801136483472281_Min_o @ B3 ) @ ( lattic1973801136483472281_Min_o @ A3 ) )
            = ( lattic1973801136483472281_Min_o @ A3 ) ) ) ) ) ).

% Min.subset
thf(fact_3088_Min_Osubset,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( B3 != bot_bot_set_nat )
       => ( ( ord_less_eq_set_nat @ B3 @ A3 )
         => ( ( ord_min_nat @ ( lattic8721135487736765967in_nat @ B3 ) @ ( lattic8721135487736765967in_nat @ A3 ) )
            = ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ).

% Min.subset
thf(fact_3089_Min_Oclosed,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ! [X2: $o,Y2: $o] : ( member_o @ ( ord_min_o @ X2 @ Y2 ) @ ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
         => ( member_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ A3 ) ) ) ) ).

% Min.closed
thf(fact_3090_Min_Oclosed,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( ord_min_nat @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
         => ( member_nat2 @ ( lattic8721135487736765967in_nat @ A3 ) @ A3 ) ) ) ) ).

% Min.closed
thf(fact_3091_Min_Oinsert__not__elem,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic1973801136483472281_Min_o @ ( insert_o @ X @ A3 ) )
            = ( ord_min_o @ X @ ( lattic1973801136483472281_Min_o @ A3 ) ) ) ) ) ) ).

% Min.insert_not_elem
thf(fact_3092_Min_Oinsert__not__elem,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ~ ( member_nat2 @ X @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic8721135487736765967in_nat @ ( insert_nat2 @ X @ A3 ) )
            = ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ A3 ) ) ) ) ) ) ).

% Min.insert_not_elem
thf(fact_3093_Min_Ounion,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( A3 != bot_bot_set_o )
       => ( ( finite_finite_o @ B3 )
         => ( ( B3 != bot_bot_set_o )
           => ( ( lattic1973801136483472281_Min_o @ ( sup_sup_set_o @ A3 @ B3 ) )
              = ( ord_min_o @ ( lattic1973801136483472281_Min_o @ A3 ) @ ( lattic1973801136483472281_Min_o @ B3 ) ) ) ) ) ) ) ).

% Min.union
thf(fact_3094_Min_Ounion,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( A3 != bot_bot_set_nat )
       => ( ( finite_finite_nat @ B3 )
         => ( ( B3 != bot_bot_set_nat )
           => ( ( lattic8721135487736765967in_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
              = ( ord_min_nat @ ( lattic8721135487736765967in_nat @ A3 ) @ ( lattic8721135487736765967in_nat @ B3 ) ) ) ) ) ) ) ).

% Min.union
thf(fact_3095_complete__linorder__inf__min,axiom,
    inf_inf_Product_unit = ord_min_Product_unit ).

% complete_linorder_inf_min
thf(fact_3096_semilattice__order__set_Osubset__imp,axiom,
    ! [F: $o > $o > $o,Less_eq: $o > $o > $o,Less: $o > $o > $o,A3: set_o,B3: set_o] :
      ( ( lattic5087519243920114290_set_o @ F @ Less_eq @ Less )
     => ( ( ord_less_eq_set_o @ A3 @ B3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( finite_finite_o @ B3 )
           => ( Less_eq @ ( lattic3100085485418696562ce_F_o @ F @ B3 ) @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_3097_semilattice__order__set_Osubset__imp,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A3: set_nat,B3: set_nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( ord_less_eq_set_nat @ A3 @ B3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( finite_finite_nat @ B3 )
           => ( Less_eq @ ( lattic7742739596368939638_F_nat @ F @ B3 ) @ ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).

% semilattice_order_set.subset_imp
thf(fact_3098_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3099_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3100_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3101_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3102_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3103_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3104_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_3105_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_3106_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_3107_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_3108_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_3109_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C2 @ A )
        = ( times_times_nat @ C2 @ B ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_3110_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C2 )
        = ( times_times_nat @ B @ C2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_3111_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3112_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_3113_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3114_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3115_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3116_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3117_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3118_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_3119_zero__comp__diff__simps_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% zero_comp_diff_simps(1)
thf(fact_3120_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_3121_zero__comp__diff__simps_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% zero_comp_diff_simps(2)
thf(fact_3122_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ( times_times_int @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_3123_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_3124_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ( times_times_int @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_3125_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_3126_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3127_min__0__1_I1_J,axiom,
    ( ( ord_min_int @ zero_zero_int @ one_one_int )
    = zero_zero_int ) ).

% min_0_1(1)
thf(fact_3128_min__0__1_I1_J,axiom,
    ( ( ord_min_nat @ zero_zero_nat @ one_one_nat )
    = zero_zero_nat ) ).

% min_0_1(1)
thf(fact_3129_min__0__1_I2_J,axiom,
    ( ( ord_min_int @ one_one_int @ zero_zero_int )
    = zero_zero_int ) ).

% min_0_1(2)
thf(fact_3130_min__0__1_I2_J,axiom,
    ( ( ord_min_nat @ one_one_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% min_0_1(2)
thf(fact_3131_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_3132_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_3133_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_3134_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_3135_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_3136_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_3137_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_3138_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_3139_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_3140_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_3141_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_3142_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_3143_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_3144_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ C2 @ A )
          = ( times_times_nat @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_3145_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C2 )
          = ( times_times_nat @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_3146_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_3147_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_3148_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z4: int] : Y5 = Z4 )
    = ( ^ [A2: int,B2: int] :
          ( ( minus_minus_int @ A2 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3149_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3150_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono
thf(fact_3151_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3152_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3153_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_3154_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3155_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3156_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3157_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3158_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3159_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3160_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3161_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_3162_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3163_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_3164_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_3165_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3166_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3167_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3168_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3169_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3170_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3171_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3172_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3173_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3174_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3175_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3176_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one_class.zero_le_one
thf(fact_3177_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% zero_less_one_class.zero_le_one
thf(fact_3178_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3179_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_int @ zero_zero_int @ one_one_int ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_3180_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_3181_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_3182_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3183_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_3184_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_3185_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_3186_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_3187_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_3188_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_3189_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_3190_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3191_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_3192_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_3193_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_3194_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_3195_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3196_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_3197_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3198_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_3199_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_3200_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_3201_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_3202_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3203_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_3204_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_3205_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_3206_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3207_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_3208_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_3209_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3210_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_3211_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_3212_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_3213_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_3214_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_3215_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_3216_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_3217_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A2: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A2 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3218_Inf__fin__def,axiom,
    ( lattic47131356835913163n_assn
    = ( lattic7606391089408249806F_assn @ inf_inf_assn ) ) ).

% Inf_fin_def
thf(fact_3219_Inf__fin__def,axiom,
    ( lattic5238388535129920115in_nat
    = ( lattic7742739596368939638_F_nat @ inf_inf_nat ) ) ).

% Inf_fin_def
thf(fact_3220_Inf__fin__def,axiom,
    ( lattic3014633134055518761et_nat
    = ( lattic4908145837437951532et_nat @ inf_inf_set_nat ) ) ).

% Inf_fin_def
thf(fact_3221_Inf__fin__def,axiom,
    ( lattic1263872656861969706t_unit
    = ( lattic2430820486025211623t_unit @ inf_inf_Product_unit ) ) ).

% Inf_fin_def
thf(fact_3222_Inf__fin__def,axiom,
    ( lattic30941717366863870at_nat
    = ( lattic2063973316643036219at_nat @ inf_in2572325071724192079at_nat ) ) ).

% Inf_fin_def
thf(fact_3223_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3224_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3225_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3226_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3227_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3228_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3229_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3230_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3231_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3232_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3233_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3234_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3235_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3236_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3237_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3238_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3239_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3240_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3241_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3242_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3243_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3244_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3245_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3246_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_3247_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3248_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3249_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3250_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3251_semilattice__order__set_OboundedE,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,Less_eq: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,Less: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( lattic7917596248206126503_nat_o @ F @ Less_eq @ Less )
     => ( ( finite3252695134891459830_nat_o @ A3 )
       => ( ( A3 != bot_bo7824918357723582233_nat_o )
         => ( ( Less_eq @ X @ ( lattic5777113317850512039_nat_o @ F @ A3 ) )
           => ! [A9: produc3658429121746597890et_nat > $o] :
                ( ( member6576561426505652726_nat_o @ A9 @ A3 )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_3252_semilattice__order__set_OboundedE,axiom,
    ! [F: $o > $o > $o,Less_eq: $o > $o > $o,Less: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic5087519243920114290_set_o @ F @ Less_eq @ Less )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( Less_eq @ X @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) )
           => ! [A9: $o] :
                ( ( member_o @ A9 @ A3 )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_3253_semilattice__order__set_OboundedE,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A3: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A3 ) )
           => ! [A9: nat] :
                ( ( member_nat2 @ A9 @ A3 )
               => ( Less_eq @ X @ A9 ) ) ) ) ) ) ).

% semilattice_order_set.boundedE
thf(fact_3254_semilattice__order__set_OboundedI,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,Less_eq: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,Less: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > $o,A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( lattic7917596248206126503_nat_o @ F @ Less_eq @ Less )
     => ( ( finite3252695134891459830_nat_o @ A3 )
       => ( ( A3 != bot_bo7824918357723582233_nat_o )
         => ( ! [A4: produc3658429121746597890et_nat > $o] :
                ( ( member6576561426505652726_nat_o @ A4 @ A3 )
               => ( Less_eq @ X @ A4 ) )
           => ( Less_eq @ X @ ( lattic5777113317850512039_nat_o @ F @ A3 ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_3255_semilattice__order__set_OboundedI,axiom,
    ! [F: $o > $o > $o,Less_eq: $o > $o > $o,Less: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic5087519243920114290_set_o @ F @ Less_eq @ Less )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ! [A4: $o] :
                ( ( member_o @ A4 @ A3 )
               => ( Less_eq @ X @ A4 ) )
           => ( Less_eq @ X @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_3256_semilattice__order__set_OboundedI,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A3: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ! [A4: nat] :
                ( ( member_nat2 @ A4 @ A3 )
               => ( Less_eq @ X @ A4 ) )
           => ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).

% semilattice_order_set.boundedI
thf(fact_3257_semilattice__order__set_Obounded__iff,axiom,
    ! [F: $o > $o > $o,Less_eq: $o > $o > $o,Less: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic5087519243920114290_set_o @ F @ Less_eq @ Less )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( Less_eq @ X @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) )
            = ( ! [X3: $o] :
                  ( ( member_o @ X3 @ A3 )
                 => ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
thf(fact_3258_semilattice__order__set_Obounded__iff,axiom,
    ! [F: nat > nat > nat,Less_eq: nat > nat > $o,Less: nat > nat > $o,A3: set_nat,X: nat] :
      ( ( lattic6009151579333465974et_nat @ F @ Less_eq @ Less )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( Less_eq @ X @ ( lattic7742739596368939638_F_nat @ F @ A3 ) )
            = ( ! [X3: nat] :
                  ( ( member_nat2 @ X3 @ A3 )
                 => ( Less_eq @ X @ X3 ) ) ) ) ) ) ) ).

% semilattice_order_set.bounded_iff
thf(fact_3259_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3260_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3261_max__mult__distrib__left,axiom,
    ! [P4: int,X: int,Y: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ P4 @ ( ord_max_int @ X @ Y ) )
          = ( ord_max_int @ ( times_times_int @ P4 @ X ) @ ( times_times_int @ P4 @ Y ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ P4 @ ( ord_max_int @ X @ Y ) )
          = ( ord_min_int @ ( times_times_int @ P4 @ X ) @ ( times_times_int @ P4 @ Y ) ) ) ) ) ).

% max_mult_distrib_left
thf(fact_3262_min__mult__distrib__left,axiom,
    ! [P4: int,X: int,Y: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ P4 @ ( ord_min_int @ X @ Y ) )
          = ( ord_min_int @ ( times_times_int @ P4 @ X ) @ ( times_times_int @ P4 @ Y ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ P4 @ ( ord_min_int @ X @ Y ) )
          = ( ord_max_int @ ( times_times_int @ P4 @ X ) @ ( times_times_int @ P4 @ Y ) ) ) ) ) ).

% min_mult_distrib_left
thf(fact_3263_max__mult__distrib__right,axiom,
    ! [P4: int,X: int,Y: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ ( ord_max_int @ X @ Y ) @ P4 )
          = ( ord_max_int @ ( times_times_int @ X @ P4 ) @ ( times_times_int @ Y @ P4 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ ( ord_max_int @ X @ Y ) @ P4 )
          = ( ord_min_int @ ( times_times_int @ X @ P4 ) @ ( times_times_int @ Y @ P4 ) ) ) ) ) ).

% max_mult_distrib_right
thf(fact_3264_min__mult__distrib__right,axiom,
    ! [P4: int,X: int,Y: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ ( ord_min_int @ X @ Y ) @ P4 )
          = ( ord_min_int @ ( times_times_int @ X @ P4 ) @ ( times_times_int @ Y @ P4 ) ) ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ P4 )
       => ( ( times_times_int @ ( ord_min_int @ X @ Y ) @ P4 )
          = ( ord_max_int @ ( times_times_int @ X @ P4 ) @ ( times_times_int @ Y @ P4 ) ) ) ) ) ).

% min_mult_distrib_right
thf(fact_3265_mult__le__cancel__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3266_mult__le__cancel__iff2,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3267_mult__less__iff1,axiom,
    ! [Z: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3268_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_a @ N @ nil_a )
          = ( cons_list_a @ nil_a @ nil_list_a ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_a @ N @ nil_a )
          = nil_list_a ) ) ) ).

% n_lists_Nil
thf(fact_3269_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_b @ N @ nil_b )
          = ( cons_list_b @ nil_b @ nil_list_b ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_b @ N @ nil_b )
          = nil_list_b ) ) ) ).

% n_lists_Nil
thf(fact_3270_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list679713369942834879n_assn @ N @ nil_Pr5671120429643327159n_assn )
          = ( cons_l2423627976422276333n_assn @ nil_Pr5671120429643327159n_assn @ nil_li5476096274760905021n_assn ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list679713369942834879n_assn @ N @ nil_Pr5671120429643327159n_assn )
          = nil_li5476096274760905021n_assn ) ) ) ).

% n_lists_Nil
thf(fact_3271_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_3272_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = ( cons_list_int @ nil_int @ nil_list_int ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_int @ N @ nil_int )
          = nil_list_int ) ) ) ).

% n_lists_Nil
thf(fact_3273_nat__geq__1__eq__neqz,axiom,
    ! [X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ X )
      = ( X != zero_zero_nat ) ) ).

% nat_geq_1_eq_neqz
thf(fact_3274_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_a] :
      ( ( n_lists_a @ zero_zero_nat @ Xs )
      = ( cons_list_a @ nil_a @ nil_list_a ) ) ).

% n_lists.simps(1)
thf(fact_3275_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_b] :
      ( ( n_lists_b @ zero_zero_nat @ Xs )
      = ( cons_list_b @ nil_b @ nil_list_b ) ) ).

% n_lists.simps(1)
thf(fact_3276_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)
thf(fact_3277_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)
thf(fact_3278_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)
thf(fact_3279_semilattice__set_Oinsert__remove,axiom,
    ! [F: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( lattic3100085485418696562ce_F_o @ F @ ( insert_o @ X @ A3 ) )
          = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                = bot_bot_set_o )
             => X )
            & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
               != bot_bot_set_o )
             => ( F @ X @ ( lattic3100085485418696562ce_F_o @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ).

% semilattice_set.insert_remove
thf(fact_3280_semilattice__set_Oinsert__remove,axiom,
    ! [F: nat > nat > nat,A3: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
              = bot_bot_set_nat )
           => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A3 ) )
              = X ) )
          & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
             != bot_bot_set_nat )
           => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A3 ) )
              = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).

% semilattice_set.insert_remove
thf(fact_3281_semilattice__set_Oremove,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( lattic767719792269653083_nat_o @ F )
     => ( ( finite3252695134891459830_nat_o @ A3 )
       => ( ( member6576561426505652726_nat_o @ X @ A3 )
         => ( ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
                = bot_bo7824918357723582233_nat_o )
             => ( ( lattic5777113317850512039_nat_o @ F @ A3 )
                = X ) )
            & ( ( ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) )
               != bot_bo7824918357723582233_nat_o )
             => ( ( lattic5777113317850512039_nat_o @ F @ A3 )
                = ( F @ X @ ( lattic5777113317850512039_nat_o @ F @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ) ) ) ).

% semilattice_set.remove
thf(fact_3282_semilattice__set_Oremove,axiom,
    ! [F: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( member_o @ X @ A3 )
         => ( ( lattic3100085485418696562ce_F_o @ F @ A3 )
            = ( ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                  = bot_bot_set_o )
               => X )
              & ( ( ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) )
                 != bot_bot_set_o )
               => ( F @ X @ ( lattic3100085485418696562ce_F_o @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ) ) ) ).

% semilattice_set.remove
thf(fact_3283_semilattice__set_Oremove,axiom,
    ! [F: nat > nat > nat,A3: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( member_nat2 @ X @ A3 )
         => ( ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
                = bot_bot_set_nat )
             => ( ( lattic7742739596368939638_F_nat @ F @ A3 )
                = X ) )
            & ( ( ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
               != bot_bot_set_nat )
             => ( ( lattic7742739596368939638_F_nat @ F @ A3 )
                = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ) ).

% semilattice_set.remove
thf(fact_3284_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_3285_semilattice__set_Ounion,axiom,
    ! [F: $o > $o > $o,A3: set_o,B3: set_o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( finite_finite_o @ B3 )
           => ( ( B3 != bot_bot_set_o )
             => ( ( lattic3100085485418696562ce_F_o @ F @ ( sup_sup_set_o @ A3 @ B3 ) )
                = ( F @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) @ ( lattic3100085485418696562ce_F_o @ F @ B3 ) ) ) ) ) ) ) ) ).

% semilattice_set.union
thf(fact_3286_semilattice__set_Ounion,axiom,
    ! [F: nat > nat > nat,A3: set_nat,B3: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( finite_finite_nat @ B3 )
           => ( ( B3 != bot_bot_set_nat )
             => ( ( lattic7742739596368939638_F_nat @ F @ ( sup_sup_set_nat @ A3 @ B3 ) )
                = ( F @ ( lattic7742739596368939638_F_nat @ F @ A3 ) @ ( lattic7742739596368939638_F_nat @ F @ B3 ) ) ) ) ) ) ) ) ).

% semilattice_set.union
thf(fact_3287_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( lattic767719792269653083_nat_o @ F )
     => ( ( finite3252695134891459830_nat_o @ A3 )
       => ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
         => ( ( A3 != bot_bo7824918357723582233_nat_o )
           => ( ( lattic5777113317850512039_nat_o @ F @ ( insert5175938949040314269_nat_o @ X @ A3 ) )
              = ( F @ X @ ( lattic5777113317850512039_nat_o @ F @ A3 ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_3288_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ~ ( member_o @ X @ A3 )
         => ( ( A3 != bot_bot_set_o )
           => ( ( lattic3100085485418696562ce_F_o @ F @ ( insert_o @ X @ A3 ) )
              = ( F @ X @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_3289_semilattice__set_Oinsert__not__elem,axiom,
    ! [F: nat > nat > nat,A3: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ~ ( member_nat2 @ X @ A3 )
         => ( ( A3 != bot_bot_set_nat )
           => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A3 ) )
              = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ) ).

% semilattice_set.insert_not_elem
thf(fact_3290_semilattice__set_Oinsert,axiom,
    ! [F: $o > $o > $o,A3: set_o,X: $o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ( lattic3100085485418696562ce_F_o @ F @ ( insert_o @ X @ A3 ) )
            = ( F @ X @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) ) ) ) ) ) ).

% semilattice_set.insert
thf(fact_3291_semilattice__set_Oinsert,axiom,
    ! [F: nat > nat > nat,A3: set_nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ A3 ) )
            = ( F @ X @ ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).

% semilattice_set.insert
thf(fact_3292_semilattice__set_Oclosed,axiom,
    ! [F: ( produc3658429121746597890et_nat > $o ) > ( produc3658429121746597890et_nat > $o ) > produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( lattic767719792269653083_nat_o @ F )
     => ( ( finite3252695134891459830_nat_o @ A3 )
       => ( ( A3 != bot_bo7824918357723582233_nat_o )
         => ( ! [X2: produc3658429121746597890et_nat > $o,Y2: produc3658429121746597890et_nat > $o] : ( member6576561426505652726_nat_o @ ( F @ X2 @ Y2 ) @ ( insert5175938949040314269_nat_o @ X2 @ ( insert5175938949040314269_nat_o @ Y2 @ bot_bo7824918357723582233_nat_o ) ) )
           => ( member6576561426505652726_nat_o @ ( lattic5777113317850512039_nat_o @ F @ A3 ) @ A3 ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_3293_semilattice__set_Oclosed,axiom,
    ! [F: $o > $o > $o,A3: set_o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( A3 != bot_bot_set_o )
         => ( ! [X2: $o,Y2: $o] : ( member_o @ ( F @ X2 @ Y2 ) @ ( insert_o @ X2 @ ( insert_o @ Y2 @ bot_bot_set_o ) ) )
           => ( member_o @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) @ A3 ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_3294_semilattice__set_Oclosed,axiom,
    ! [F: nat > nat > nat,A3: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( A3 != bot_bot_set_nat )
         => ( ! [X2: nat,Y2: nat] : ( member_nat2 @ ( F @ X2 @ Y2 ) @ ( insert_nat2 @ X2 @ ( insert_nat2 @ Y2 @ bot_bot_set_nat ) ) )
           => ( member_nat2 @ ( lattic7742739596368939638_F_nat @ F @ A3 ) @ A3 ) ) ) ) ) ).

% semilattice_set.closed
thf(fact_3295_semilattice__set_Osubset,axiom,
    ! [F: $o > $o > $o,A3: set_o,B3: set_o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( finite_finite_o @ A3 )
       => ( ( B3 != bot_bot_set_o )
         => ( ( ord_less_eq_set_o @ B3 @ A3 )
           => ( ( F @ ( lattic3100085485418696562ce_F_o @ F @ B3 ) @ ( lattic3100085485418696562ce_F_o @ F @ A3 ) )
              = ( lattic3100085485418696562ce_F_o @ F @ A3 ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_3296_semilattice__set_Osubset,axiom,
    ! [F: nat > nat > nat,A3: set_nat,B3: set_nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( finite_finite_nat @ A3 )
       => ( ( B3 != bot_bot_set_nat )
         => ( ( ord_less_eq_set_nat @ B3 @ A3 )
           => ( ( F @ ( lattic7742739596368939638_F_nat @ F @ B3 ) @ ( lattic7742739596368939638_F_nat @ F @ A3 ) )
              = ( lattic7742739596368939638_F_nat @ F @ A3 ) ) ) ) ) ) ).

% semilattice_set.subset
thf(fact_3297_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3298_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_3299_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_3300_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_3301_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_3302_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_3303_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_3304_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_3305_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_3306_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_3307_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_3308_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_3309_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_3310_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_3311_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_3312_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_3313_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_3314_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_3315_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_3316_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_3317_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_3318_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_3319_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_3320_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_3321_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3322_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_3323_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_3324_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_3325_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3326_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_3327_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3328_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_3329_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3330_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_3331_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3332_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_3333_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3334_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3335_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_3336_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_3337_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_3338_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_3339_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_3340_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_3341_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_3342_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_3343_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_3344_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_3345_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_3346_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_3347_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_3348_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_3349_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_3350_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_3351_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_3352_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_3353_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_3354_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_3355_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_3356_exists__leI,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [N5: nat] :
            ( ( ord_less_nat @ N5 @ N )
           => ~ ( P @ N5 ) )
       => ( P @ N ) )
     => ? [N6: nat] :
          ( ( ord_less_eq_nat @ N6 @ N )
          & ( P @ N6 ) ) ) ).

% exists_leI
thf(fact_3357_add_Oright__commute,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ).

% add.right_commute
thf(fact_3358_add_Oright__commute,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% add.right_commute
thf(fact_3359_add_Oright__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.right_assoc
thf(fact_3360_add_Oright__assoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.right_assoc
thf(fact_3361_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_3362_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_3363_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_3364_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_3365_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3366_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add.left_commute
thf(fact_3367_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3368_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A2: int,B2: int] : ( plus_plus_int @ B2 @ A2 ) ) ) ).

% ab_semigroup_add_class.add.commute
thf(fact_3369_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_3370_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_3371_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_3372_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_3373_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3374_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_3375_group__cancel_Oadd1,axiom,
    ! [A3: nat,K: nat,A: nat,B: nat] :
      ( ( A3
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A3 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3376_group__cancel_Oadd1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_3377_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3378_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_3379_mlex__leI,axiom,
    ! [A: nat,A8: nat,B: nat,B8: nat,N4: nat] :
      ( ( ord_less_eq_nat @ A @ A8 )
     => ( ( ord_less_eq_nat @ B @ B8 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N4 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A8 @ N4 ) @ B8 ) ) ) ) ).

% mlex_leI
thf(fact_3380_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X3: int] : ( plus_plus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_3381_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3382_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_3383_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3384_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_3385_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A2: nat,B2: nat] :
        ? [C5: nat] :
          ( B2
          = ( plus_plus_nat @ A2 @ C5 ) ) ) ) ).

% le_iff_add
thf(fact_3386_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_3387_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_3388_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( plus_plus_nat @ A @ C ) ) ) ).

% less_eqE
thf(fact_3389_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_3390_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_3391_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_3392_add__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_mono
thf(fact_3393_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3394_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_3395_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3396_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_3397_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3398_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_3399_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3400_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_3401_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_3402_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3403_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_3404_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3405_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_3406_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3407_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_3408_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_3409_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_3410_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3411_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_3412_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3413_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_3414_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3415_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_3416_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3417_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_3418_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3419_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_3420_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_3421_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_3422_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3423_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_3424_distrib__left,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_3425_distrib__left,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_3426_distrib__right,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_3427_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_3428_combine__common__factor,axiom,
    ! [A: int,E3: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E3 ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3429_combine__common__factor,axiom,
    ! [A: nat,E3: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E3 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E3 ) @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E3 ) @ C2 ) ) ).

% combine_common_factor
thf(fact_3430_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3431_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_3432_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3433_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_3434_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_3435_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_3436_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_3437_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_3438_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C2 @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_3439_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_3440_group__cancel_Osub1,axiom,
    ! [A3: int,K: int,A: int,B: int] :
      ( ( A3
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A3 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_3441_min__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_min_int @ Y @ Z ) )
      = ( ord_min_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_3442_min__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_min_nat @ Y @ Z ) )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% min_add_distrib_right
thf(fact_3443_min__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_min_int @ X @ Y ) @ Z )
      = ( ord_min_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_3444_min__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_min_nat @ X @ Y ) @ Z )
      = ( ord_min_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% min_add_distrib_left
thf(fact_3445_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3446_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z ) ) ) ).

% max_add_distrib_right
thf(fact_3447_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z ) @ ( plus_plus_nat @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3448_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z ) @ ( plus_plus_int @ Y @ Z ) ) ) ).

% max_add_distrib_left
thf(fact_3449_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic7983604933768736026t_assn @ inf_inf_assn ).

% Inf_fin.semilattice_set_axioms
thf(fact_3450_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic1029310888574255042et_nat @ inf_inf_nat ).

% Inf_fin.semilattice_set_axioms
thf(fact_3451_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic6452893353811829624et_nat @ inf_inf_set_nat ).

% Inf_fin.semilattice_set_axioms
thf(fact_3452_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic356177490429882523t_unit @ inf_inf_Product_unit ).

% Inf_fin.semilattice_set_axioms
thf(fact_3453_Inf__fin_Osemilattice__set__axioms,axiom,
    lattic6581507972311975919at_nat @ inf_in2572325071724192079at_nat ).

% Inf_fin.semilattice_set_axioms
thf(fact_3454_add_Osafe__commute,axiom,
    ! [X: nat,Y: nat,A: nat,B: nat] :
      ( ( syntax4682126007086162916at_nat @ ( plus_plus_nat @ X @ Y ) @ A )
     => ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_3455_add_Osafe__commute,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( syntax5678989248478167196nt_int @ ( plus_plus_int @ X @ Y ) @ A )
     => ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ B @ A ) ) ) ).

% add.safe_commute
thf(fact_3456_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_3457_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_3458_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3459_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_3460_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3461_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3462_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3463_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_3464_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3465_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3466_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3467_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3468_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3469_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3470_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3471_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3472_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3473_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_less_le_mono
thf(fact_3474_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3475_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D2 )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D2 ) ) ) ) ).

% add_le_less_mono
thf(fact_3476_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3477_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_3478_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3479_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_3480_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3481_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_3482_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3483_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3484_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3485_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3486_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3487_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_3488_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_3489_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_3490_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% le_add_diff
thf(fact_3491_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_3492_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_3493_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_3494_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_3495_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_3496_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_3497_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_3498_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C2 )
          = ( B
            = ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_3499_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_3500_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_3501_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_3502_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_3503_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_3504_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_3505_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_3506_eq__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( C2
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E3 ) @ D2 ) ) ) ).

% eq_add_iff2
thf(fact_3507_eq__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E3 ) @ C2 )
        = D2 ) ) ).

% eq_add_iff1
thf(fact_3508_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3509_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_3510_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3511_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_3512_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3513_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3514_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3515_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3516_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3517_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3518_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3519_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3520_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3521_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3522_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3523_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3524_le__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E3 ) @ D2 ) ) ) ).

% le_add_iff2
thf(fact_3525_le__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E3 ) @ C2 ) @ D2 ) ) ).

% le_add_iff1
thf(fact_3526_less__add__iff1,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E3 ) @ C2 ) @ D2 ) ) ).

% less_add_iff1
thf(fact_3527_less__add__iff2,axiom,
    ! [A: int,E3: int,C2: int,B: int,D2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E3 ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E3 ) @ D2 ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E3 ) @ D2 ) ) ) ).

% less_add_iff2
thf(fact_3528_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_3529_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3530_semilattice__set_Osingleton,axiom,
    ! [F: $o > $o > $o,X: $o] :
      ( ( lattic7139874143898644262_set_o @ F )
     => ( ( lattic3100085485418696562ce_F_o @ F @ ( insert_o @ X @ bot_bot_set_o ) )
        = X ) ) ).

% semilattice_set.singleton
thf(fact_3531_semilattice__set_Osingleton,axiom,
    ! [F: nat > nat > nat,X: nat] :
      ( ( lattic1029310888574255042et_nat @ F )
     => ( ( lattic7742739596368939638_F_nat @ F @ ( insert_nat2 @ X @ bot_bot_set_nat ) )
        = X ) ) ).

% semilattice_set.singleton
thf(fact_3532_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3533_discrete,axiom,
    ( ord_less_int
    = ( ^ [A2: int] : ( ord_less_eq_int @ ( plus_plus_int @ A2 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3534_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3535_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3536_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_3537_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C2: int,D2: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C2 ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3538_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C2: nat,D2: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D2 ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C2 ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D2 ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3539_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D2: int] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D2 ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3540_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D2: nat] :
      ( ( ( A != B )
        & ( C2 != D2 ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D2 ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_3541_crossproduct__eq,axiom,
    ! [W: int,Y: int,X: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3542_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W = X )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_3543_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X3: int] : ( minus_minus_int @ ( plus_plus_int @ X3 @ X3 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_3544_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: nat,Xs: list_nat] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( cons_nat @ X @ ( slice_nat @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_nat @ Begin @ End @ ( cons_nat @ X @ Xs ) )
          = ( slice_nat @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_3545_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: int,Xs: list_int] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( cons_int @ X @ ( slice_int @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_int @ Begin @ End @ ( cons_int @ X @ Xs ) )
          = ( slice_int @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_3546_slice__Cons,axiom,
    ! [Begin: nat,End: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( ( Begin = zero_zero_nat )
          & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_7964966981652229398n_assn @ Begin @ End @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( cons_P2971678138204555879n_assn @ X @ ( slice_7964966981652229398n_assn @ Begin @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) )
      & ( ~ ( ( Begin = zero_zero_nat )
            & ( ord_less_nat @ zero_zero_nat @ End ) )
       => ( ( slice_7964966981652229398n_assn @ Begin @ End @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
          = ( slice_7964966981652229398n_assn @ ( minus_minus_nat @ Begin @ one_one_nat ) @ ( minus_minus_nat @ End @ one_one_nat ) @ Xs ) ) ) ) ).

% slice_Cons
thf(fact_3547_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_a @ Begin @ End @ nil_a )
      = nil_a ) ).

% slice_Nil
thf(fact_3548_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_b @ Begin @ End @ nil_b )
      = nil_b ) ).

% slice_Nil
thf(fact_3549_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_7964966981652229398n_assn @ Begin @ End @ nil_Pr5671120429643327159n_assn )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_Nil
thf(fact_3550_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_nat @ Begin @ End @ nil_nat )
      = nil_nat ) ).

% slice_Nil
thf(fact_3551_slice__Nil,axiom,
    ! [Begin: nat,End: nat] :
      ( ( slice_int @ Begin @ End @ nil_int )
      = nil_int ) ).

% slice_Nil
thf(fact_3552_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_a] :
      ( ( slice_a @ I @ I @ Xs )
      = nil_a ) ).

% slice_eq_bounds_empty
thf(fact_3553_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_b] :
      ( ( slice_b @ I @ I @ Xs )
      = nil_b ) ).

% slice_eq_bounds_empty
thf(fact_3554_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( slice_7964966981652229398n_assn @ I @ I @ Xs )
      = nil_Pr5671120429643327159n_assn ) ).

% slice_eq_bounds_empty
thf(fact_3555_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_nat] :
      ( ( slice_nat @ I @ I @ Xs )
      = nil_nat ) ).

% slice_eq_bounds_empty
thf(fact_3556_slice__eq__bounds__empty,axiom,
    ! [I: nat,Xs: list_int] :
      ( ( slice_int @ I @ I @ Xs )
      = nil_int ) ).

% slice_eq_bounds_empty
thf(fact_3557_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_3558_mlex__snd__decrI,axiom,
    ! [A: nat,A8: nat,B: nat,B8: nat,N4: nat] :
      ( ( A = A8 )
     => ( ( ord_less_nat @ B @ B8 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N4 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A8 @ N4 ) @ B8 ) ) ) ) ).

% mlex_snd_decrI
thf(fact_3559_mlex__fst__decrI,axiom,
    ! [A: nat,A8: nat,B: nat,N4: nat,B8: nat] :
      ( ( ord_less_nat @ A @ A8 )
     => ( ( ord_less_nat @ B @ N4 )
       => ( ( ord_less_nat @ B8 @ N4 )
         => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N4 ) @ B ) @ ( plus_plus_nat @ ( times_times_nat @ A8 @ N4 ) @ B8 ) ) ) ) ) ).

% mlex_fst_decrI
thf(fact_3560_mlex__bound,axiom,
    ! [A: nat,A3: nat,B: nat,N4: nat] :
      ( ( ord_less_nat @ A @ A3 )
     => ( ( ord_less_nat @ B @ N4 )
       => ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ A @ N4 ) @ B ) @ ( times_times_nat @ A3 @ N4 ) ) ) ) ).

% mlex_bound
thf(fact_3561_count__list_Osimps_I2_J,axiom,
    ! [X: nat,Y: nat,Xs: list_nat] :
      ( ( ( X = Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_nat @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_nat @ ( cons_nat @ X @ Xs ) @ Y )
          = ( count_list_nat @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_3562_count__list_Osimps_I2_J,axiom,
    ! [X: int,Y: int,Xs: list_int] :
      ( ( ( X = Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_list_int @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_list_int @ ( cons_int @ X @ Xs ) @ Y )
          = ( count_list_int @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_3563_count__list_Osimps_I2_J,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( X = Y )
       => ( ( count_2530312006313534765n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Y )
          = ( plus_plus_nat @ ( count_2530312006313534765n_assn @ Xs @ Y ) @ one_one_nat ) ) )
      & ( ( X != Y )
       => ( ( count_2530312006313534765n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ Y )
          = ( count_2530312006313534765n_assn @ Xs @ Y ) ) ) ) ).

% count_list.simps(2)
thf(fact_3564_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3565_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_3566_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_3567_nth__Cons__pos,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_3568_nth__Cons__pos,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
        = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_3569_power__minus__mult,axiom,
    ! [N: nat,A: assn] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_assn @ ( power_power_assn @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_assn @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3570_power__minus__mult,axiom,
    ! [N: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
        = ( power_power_nat @ A @ N ) ) ) ).

% power_minus_mult
thf(fact_3571_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M3: nat] : ( if_int @ ( M3 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3572_power__eq__if,axiom,
    ( power_power_assn
    = ( ^ [P6: assn,M3: nat] : ( if_assn @ ( M3 = zero_zero_nat ) @ one_one_assn @ ( times_times_assn @ P6 @ ( power_power_assn @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3573_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M3 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3574_card__Diff__singleton,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( member6576561426505652726_nat_o @ X @ A3 )
     => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
        = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3575_card__Diff__singleton,axiom,
    ! [X: $o,A3: set_o] :
      ( ( member_o @ X @ A3 )
     => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) )
        = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3576_card__Diff__singleton,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( member_nat2 @ X @ A3 )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).

% card_Diff_singleton
thf(fact_3577_card__Diff__singleton__if,axiom,
    ! [X: produc3658429121746597890et_nat > $o,A3: set_Pr4532377907799695533_nat_o] :
      ( ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
          = ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) )
          = ( finite1363419556375932405_nat_o @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3578_card__Diff__singleton__if,axiom,
    ! [X: $o,A3: set_o] :
      ( ( ( member_o @ X @ A3 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_o @ X @ A3 )
       => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) )
          = ( finite_card_o @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3579_card__Diff__singleton__if,axiom,
    ! [X: nat,A3: set_nat] :
      ( ( ( member_nat2 @ X @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member_nat2 @ X @ A3 )
       => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_Diff_singleton_if
thf(fact_3580_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_assn @ one_one_assn @ N )
      = one_one_assn ) ).

% power_one
thf(fact_3581_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_3582_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_3583_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3584_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_3585_card_Oempty,axiom,
    ( ( finite_card_o @ bot_bot_set_o )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3586_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_3587_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_3588_nth__Cons__0,axiom,
    ! [X: int,Xs: list_int] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_3589_nth__Cons__0,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_3590_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_3591_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_3592_power__strict__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3593_power__strict__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_3594_card__0__eq,axiom,
    ! [A3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( ( finite_card_o @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_o ) ) ) ).

% card_0_eq
thf(fact_3595_card__0__eq,axiom,
    ! [A3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( ( finite_card_nat @ A3 )
          = zero_zero_nat )
        = ( A3 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_3596_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3597_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_3598_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3599_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_3600_power__commuting__commutes,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = ( times_times_assn @ Y @ X ) )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ Y )
        = ( times_times_assn @ Y @ ( power_power_assn @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3601_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_3602_power__mult__distrib,axiom,
    ! [A: assn,B: assn,N: nat] :
      ( ( power_power_assn @ ( times_times_assn @ A @ B ) @ N )
      = ( times_times_assn @ ( power_power_assn @ A @ N ) @ ( power_power_assn @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_3603_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_3604_power__commutes,axiom,
    ! [A: assn,N: nat] :
      ( ( times_times_assn @ ( power_power_assn @ A @ N ) @ A )
      = ( times_times_assn @ A @ ( power_power_assn @ A @ N ) ) ) ).

% power_commutes
thf(fact_3605_power__commutes,axiom,
    ! [A: nat,N: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).

% power_commutes
thf(fact_3606_map__eq__nth__eq,axiom,
    ! [F: nat > nat,L: list_nat,L3: list_nat,I: nat] :
      ( ( ( map_nat_nat @ F @ L )
        = ( map_nat_nat @ F @ L3 ) )
     => ( ( F @ ( nth_nat @ L @ I ) )
        = ( F @ ( nth_nat @ L3 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_3607_map__eq__nth__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,L: list_P8527749157015355191n_assn,L3: list_P8527749157015355191n_assn,I: nat] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ L )
        = ( map_Pr8991440229025900053n_assn @ F @ L3 ) )
     => ( ( F @ ( nth_Pr1769885009046257848n_assn @ L @ I ) )
        = ( F @ ( nth_Pr1769885009046257848n_assn @ L3 @ I ) ) ) ) ).

% map_eq_nth_eq
thf(fact_3608_one__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).

% one_le_power
thf(fact_3609_one__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).

% one_le_power
thf(fact_3610_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3611_left__right__inverse__power,axiom,
    ! [X: assn,Y: assn,N: nat] :
      ( ( ( times_times_assn @ X @ Y )
        = one_one_assn )
     => ( ( times_times_assn @ ( power_power_assn @ X @ N ) @ ( power_power_assn @ Y @ N ) )
        = one_one_assn ) ) ).

% left_right_inverse_power
thf(fact_3612_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3613_power__0,axiom,
    ! [A: assn] :
      ( ( power_power_assn @ A @ zero_zero_nat )
      = one_one_assn ) ).

% power_0
thf(fact_3614_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3615_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3616_power__add,axiom,
    ! [A: assn,M2: nat,N: nat] :
      ( ( power_power_assn @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_assn @ ( power_power_assn @ A @ M2 ) @ ( power_power_assn @ A @ N ) ) ) ).

% power_add
thf(fact_3617_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_3618_card__eq__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_o )
        | ~ ( finite_finite_o @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_3619_card__eq__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = zero_zero_nat )
      = ( ( A3 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A3 ) ) ) ).

% card_eq_0_iff
thf(fact_3620_power__le__one,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ A @ one_one_nat )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).

% power_le_one
thf(fact_3621_power__le__one,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ A @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).

% power_le_one
thf(fact_3622_power__less__power__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3623_power__less__power__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_3624_power__gt1__lemma,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3625_power__gt1__lemma,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_3626_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% power_0_left
thf(fact_3627_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_power_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% power_0_left
thf(fact_3628_card__1__singletonE,axiom,
    ! [A3: set_o] :
      ( ( ( finite_card_o @ A3 )
        = one_one_nat )
     => ~ ! [X2: $o] :
            ( A3
           != ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).

% card_1_singletonE
thf(fact_3629_card__1__singletonE,axiom,
    ! [A3: set_nat] :
      ( ( ( finite_card_nat @ A3 )
        = one_one_nat )
     => ~ ! [X2: nat] :
            ( A3
           != ( insert_nat2 @ X2 @ bot_bot_set_nat ) ) ) ).

% card_1_singletonE
thf(fact_3630_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3631_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_3632_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_3633_power__strict__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_3634_power__increasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_3635_power__increasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_3636_card__gt__0__iff,axiom,
    ! [A3: set_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
      = ( ( A3 != bot_bot_set_o )
        & ( finite_finite_o @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_3637_card__gt__0__iff,axiom,
    ! [A3: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
      = ( ( A3 != bot_bot_set_nat )
        & ( finite_finite_nat @ A3 ) ) ) ).

% card_gt_0_iff
thf(fact_3638_power__Suc__less,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ one_one_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3639_power__Suc__less,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ one_one_int )
       => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_3640_card__1__singletonI,axiom,
    ! [S: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ S )
     => ( ( ( finite1363419556375932405_nat_o @ S )
          = one_one_nat )
       => ( ( member6576561426505652726_nat_o @ X @ S )
         => ( S
            = ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_3641_card__1__singletonI,axiom,
    ! [S: set_o,X: $o] :
      ( ( finite_finite_o @ S )
     => ( ( ( finite_card_o @ S )
          = one_one_nat )
       => ( ( member_o @ X @ S )
         => ( S
            = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ).

% card_1_singletonI
thf(fact_3642_card__1__singletonI,axiom,
    ! [S: set_nat,X: nat] :
      ( ( finite_finite_nat @ S )
     => ( ( ( finite_card_nat @ S )
          = one_one_nat )
       => ( ( member_nat2 @ X @ S )
         => ( S
            = ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ).

% card_1_singletonI
thf(fact_3643_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3644_power__strict__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N @ N4 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3645_card__Diff1__le,axiom,
    ! [A3: set_o,X: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ).

% card_Diff1_le
thf(fact_3646_card__Diff1__le,axiom,
    ! [A3: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ).

% card_Diff1_le
thf(fact_3647_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3648_power__decreasing,axiom,
    ! [N: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3649_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3650_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_3651_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_3652_nth__Cons_H,axiom,
    ! [N: nat,X: int,Xs: list_int] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_3653_nth__Cons_H,axiom,
    ! [N: nat,X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_3654_self__le__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3655_self__le__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_eq_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_3656_card__Un__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) )
          = ( plus_plus_nat @ ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_3657_card__Un__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) )
          = ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B3 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ) ).

% card_Un_Int
thf(fact_3658_one__less__power,axiom,
    ! [A: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3659_one__less__power,axiom,
    ! [A: int,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_3660_card__Diff__subset__Int,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B3 ) )
     => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B3 ) )
        = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B3 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_3661_card__Diff__subset__Int,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) )
     => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B3 ) )
        = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B3 ) ) ) ) ) ).

% card_Diff_subset_Int
thf(fact_3662_count__list_Osimps_I1_J,axiom,
    ! [Y: a] :
      ( ( count_list_a @ nil_a @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3663_count__list_Osimps_I1_J,axiom,
    ! [Y: b] :
      ( ( count_list_b @ nil_b @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3664_count__list_Osimps_I1_J,axiom,
    ! [Y: produc6575502325842934193n_assn] :
      ( ( count_2530312006313534765n_assn @ nil_Pr5671120429643327159n_assn @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3665_count__list_Osimps_I1_J,axiom,
    ! [Y: nat] :
      ( ( count_list_nat @ nil_nat @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3666_count__list_Osimps_I1_J,axiom,
    ! [Y: int] :
      ( ( count_list_int @ nil_int @ Y )
      = zero_zero_nat ) ).

% count_list.simps(1)
thf(fact_3667_card__Diff1__less__iff,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A3 ) )
      = ( ( finite3252695134891459830_nat_o @ A3 )
        & ( member6576561426505652726_nat_o @ X @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3668_card__Diff1__less__iff,axiom,
    ! [A3: set_o,X: $o] :
      ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) )
      = ( ( finite_finite_o @ A3 )
        & ( member_o @ X @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3669_card__Diff1__less__iff,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) )
      = ( ( finite_finite_nat @ A3 )
        & ( member_nat2 @ X @ A3 ) ) ) ).

% card_Diff1_less_iff
thf(fact_3670_card__Diff2__less,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( member6576561426505652726_nat_o @ Y @ A3 )
         => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) @ ( insert5175938949040314269_nat_o @ Y @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3671_card__Diff2__less,axiom,
    ! [A3: set_o,X: $o,Y: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( member_o @ Y @ A3 )
         => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) @ ( insert_o @ Y @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3672_card__Diff2__less,axiom,
    ! [A3: set_nat,X: nat,Y: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( member_nat2 @ Y @ A3 )
         => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) @ ( insert_nat2 @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ) ).

% card_Diff2_less
thf(fact_3673_card__Diff1__less,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ord_less_nat @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) @ ( finite1363419556375932405_nat_o @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_3674_card__Diff1__less,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_3675_card__Diff1__less,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ).

% card_Diff1_less
thf(fact_3676_card__Un__disjoint,axiom,
    ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A3 )
     => ( ( finite6177210948735845034at_nat @ B3 )
       => ( ( ( inf_in2572325071724192079at_nat @ A3 @ B3 )
            = bot_bo2099793752762293965at_nat )
         => ( ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B3 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_3677_card__Un__disjoint,axiom,
    ! [A3: set_o,B3: set_o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_finite_o @ B3 )
       => ( ( ( inf_inf_set_o @ A3 @ B3 )
            = bot_bot_set_o )
         => ( ( finite_card_o @ ( sup_sup_set_o @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B3 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_3678_card__Un__disjoint,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_finite_nat @ B3 )
       => ( ( ( inf_inf_set_nat @ A3 @ B3 )
            = bot_bot_set_nat )
         => ( ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B3 ) )
            = ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B3 ) ) ) ) ) ) ).

% card_Un_disjoint
thf(fact_3679_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_3680_nth__non__equal__first__eq,axiom,
    ! [X: int,Y: int,Xs: list_int,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_3681_nth__non__equal__first__eq,axiom,
    ! [X: produc6575502325842934193n_assn,Y: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ N )
          = Y )
        = ( ( ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_3682_foldr__max__sorted,axiom,
    ! [Xs: list_nat,Y: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
     => ( ( ( Xs = nil_nat )
         => ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y )
            = Y ) )
        & ( ( Xs != nil_nat )
         => ( ( foldr_nat_nat @ ord_max_nat @ Xs @ Y )
            = ( ord_max_nat @ ( nth_nat @ Xs @ zero_zero_nat ) @ Y ) ) ) ) ) ).

% foldr_max_sorted
thf(fact_3683_foldr__max__sorted,axiom,
    ! [Xs: list_int,Y: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
     => ( ( ( Xs = nil_int )
         => ( ( foldr_int_int @ ord_max_int @ Xs @ Y )
            = Y ) )
        & ( ( Xs != nil_int )
         => ( ( foldr_int_int @ ord_max_int @ Xs @ Y )
            = ( ord_max_int @ ( nth_int @ Xs @ zero_zero_nat ) @ Y ) ) ) ) ) ).

% foldr_max_sorted
thf(fact_3684_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_nat,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_nat @ ( slice_nat @ From @ To @ Xs ) @ I )
            = ( nth_nat @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3685_slice__nth,axiom,
    ! [From: nat,To: nat,Xs: list_int,I: nat] :
      ( ( ord_less_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( ord_less_nat @ I @ ( minus_minus_nat @ To @ From ) )
         => ( ( nth_int @ ( slice_int @ From @ To @ Xs ) @ I )
            = ( nth_int @ Xs @ ( plus_plus_nat @ From @ I ) ) ) ) ) ) ).

% slice_nth
thf(fact_3686_last__conv__nth,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ( ( last_a @ Xs )
        = ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_3687_last__conv__nth,axiom,
    ! [Xs: list_b] :
      ( ( Xs != nil_b )
     => ( ( last_b @ Xs )
        = ( nth_b @ Xs @ ( minus_minus_nat @ ( size_size_list_b @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_3688_last__conv__nth,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( Xs != nil_Pr5671120429643327159n_assn )
     => ( ( last_P8723976779861936080n_assn @ Xs )
        = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ ( size_s6829681357464350627n_assn @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_3689_last__conv__nth,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ Xs )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_3690_last__conv__nth,axiom,
    ! [Xs: list_int] :
      ( ( Xs != nil_int )
     => ( ( last_int @ Xs )
        = ( nth_int @ Xs @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_3691_nth__Cons__numeral,axiom,
    ! [X: nat,Xs: list_nat,V: num] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_3692_nth__Cons__numeral,axiom,
    ! [X: int,Xs: list_int,V: num] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_int @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_3693_nth__Cons__numeral,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,V: num] :
      ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_Pr1769885009046257848n_assn @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_3694_card__insert__disjoint_H,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ~ ( member_o @ X @ A3 )
       => ( ( minus_minus_nat @ ( finite_card_o @ ( insert_o @ X @ A3 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite_card_o @ A3 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_3695_card__insert__disjoint_H,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ~ ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( minus_minus_nat @ ( finite1363419556375932405_nat_o @ ( insert5175938949040314269_nat_o @ X @ A3 ) ) @ ( suc @ zero_zero_nat ) )
          = ( finite1363419556375932405_nat_o @ A3 ) ) ) ) ).

% card_insert_disjoint'
thf(fact_3696_card_Oremove,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( finite1363419556375932405_nat_o @ A3 )
          = ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3697_card_Oremove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( finite_card_o @ A3 )
          = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3698_card_Oremove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( finite_card_nat @ A3 )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_3699_card_Oinsert__remove,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( finite_card_o @ ( insert_o @ X @ A3 ) )
        = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3700_card_Oinsert__remove,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( finite_card_nat @ ( insert_nat2 @ X @ A3 ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_3701_card__Suc__Diff1,axiom,
    ! [A3: set_Pr4532377907799695533_nat_o,X: produc3658429121746597890et_nat > $o] :
      ( ( finite3252695134891459830_nat_o @ A3 )
     => ( ( member6576561426505652726_nat_o @ X @ A3 )
       => ( ( suc @ ( finite1363419556375932405_nat_o @ ( minus_1801376950450012436_nat_o @ A3 @ ( insert5175938949040314269_nat_o @ X @ bot_bo7824918357723582233_nat_o ) ) ) )
          = ( finite1363419556375932405_nat_o @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3702_card__Suc__Diff1,axiom,
    ! [A3: set_o,X: $o] :
      ( ( finite_finite_o @ A3 )
     => ( ( member_o @ X @ A3 )
       => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
          = ( finite_card_o @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3703_card__Suc__Diff1,axiom,
    ! [A3: set_nat,X: nat] :
      ( ( finite_finite_nat @ A3 )
     => ( ( member_nat2 @ X @ A3 )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat2 @ X @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A3 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_3704_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3705_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3706_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3707_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3708_min__Suc__gt_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ ( suc @ A ) @ B )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(1)
thf(fact_3709_min__Suc__gt_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_min_nat @ B @ ( suc @ A ) )
        = ( suc @ A ) ) ) ).

% min_Suc_gt(2)
thf(fact_3710_length__map,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn] :
      ( ( size_size_list_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) )
      = ( size_s6829681357464350627n_assn @ Xs ) ) ).

% length_map
thf(fact_3711_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_3712_length__map,axiom,
    ! [F: int > nat,Xs: list_int] :
      ( ( size_size_list_nat @ ( map_int_nat @ F @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_3713_length__map,axiom,
    ! [F: nat > int,Xs: list_nat] :
      ( ( size_size_list_int @ ( map_nat_int @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_3714_length__map,axiom,
    ! [F: int > int,Xs: list_int] :
      ( ( size_size_list_int @ ( map_int_int @ F @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_map
thf(fact_3715_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us2: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us2 )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us2 )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_3716_append__eq__append__conv,axiom,
    ! [Xs: list_int,Ys: list_int,Us2: list_int,Vs: list_int] :
      ( ( ( ( size_size_list_int @ Xs )
          = ( size_size_list_int @ Ys ) )
        | ( ( size_size_list_int @ Us2 )
          = ( size_size_list_int @ Vs ) ) )
     => ( ( ( append_int @ Xs @ Us2 )
          = ( append_int @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_3717_length__rev,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rev
thf(fact_3718_length__rev,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rev_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rev
thf(fact_3719_length__rotate1,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rotate1
thf(fact_3720_length__rotate1,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( rotate1_int @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_rotate1
thf(fact_3721_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3722_distrib__left__numeral,axiom,
    ! [V: num,B: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3723_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3724_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_3725_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3726_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3727_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_a ) ) ).

% length_0_conv
thf(fact_3728_length__0__conv,axiom,
    ! [Xs: list_b] :
      ( ( ( size_size_list_b @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_b ) ) ).

% length_0_conv
thf(fact_3729_length__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr5671120429643327159n_assn ) ) ).

% length_0_conv
thf(fact_3730_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_3731_length__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_int ) ) ).

% length_0_conv
thf(fact_3732_power__add__numeral2,axiom,
    ! [A: assn,M2: num,N: num,B: assn] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3733_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_3734_power__add__numeral,axiom,
    ! [A: assn,M2: num,N: num] :
      ( ( times_times_assn @ ( power_power_assn @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_assn @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_assn @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3735_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_3736_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = one_one_nat ) ).

% min_0_1(6)
thf(fact_3737_min__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = one_one_int ) ).

% min_0_1(6)
thf(fact_3738_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = one_one_nat ) ).

% min_0_1(5)
thf(fact_3739_min__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_min_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = one_one_int ) ).

% min_0_1(5)
thf(fact_3740_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_3741_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_3742_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_3743_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_3744_nth__Cons__Suc,axiom,
    ! [X: nat,Xs: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_nat @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_3745_nth__Cons__Suc,axiom,
    ! [X: int,Xs: list_int,N: nat] :
      ( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_int @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_3746_nth__Cons__Suc,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( nth_Pr1769885009046257848n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) @ ( suc @ N ) )
      = ( nth_Pr1769885009046257848n_assn @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_3747_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_3748_length__append,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_size_list_int @ ( append_int @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_append
thf(fact_3749_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A3: set_int] :
      ( ( size_size_list_int @ ( linord2612477271533052124et_int @ A3 ) )
      = ( finite_card_int @ A3 ) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_3750_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] :
      ( ( size_size_list_nat @ ( linord2614967742042102400et_nat @ A3 ) )
      = ( finite_card_nat @ A3 ) ) ).

% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_3751_slice__complete,axiom,
    ! [Xs: list_nat] :
      ( ( slice_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3752_slice__complete,axiom,
    ! [Xs: list_int] :
      ( ( slice_int @ zero_zero_nat @ ( size_size_list_int @ Xs ) @ Xs )
      = Xs ) ).

% slice_complete
thf(fact_3753_length__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( product_nat_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3754_length__product,axiom,
    ! [Xs: list_nat,Ys: list_int] :
      ( ( size_s2970893825323803983at_int @ ( product_nat_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3755_length__product,axiom,
    ! [Xs: list_int,Ys: list_nat] :
      ( ( size_s7647898544948552527nt_nat @ ( product_int_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3756_length__product,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( size_s5157815400016825771nt_int @ ( product_int_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3757_length__greater__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
      = ( Xs != nil_a ) ) ).

% length_greater_0_conv
thf(fact_3758_length__greater__0__conv,axiom,
    ! [Xs: list_b] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_b @ Xs ) )
      = ( Xs != nil_b ) ) ).

% length_greater_0_conv
thf(fact_3759_length__greater__0__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s6829681357464350627n_assn @ Xs ) )
      = ( Xs != nil_Pr5671120429643327159n_assn ) ) ).

% length_greater_0_conv
thf(fact_3760_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_3761_length__greater__0__conv,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
      = ( Xs != nil_int ) ) ).

% length_greater_0_conv
thf(fact_3762_length__concat__rev,axiom,
    ! [Xs: list_list_nat] :
      ( ( size_size_list_nat @ ( concat_nat @ ( rev_list_nat @ Xs ) ) )
      = ( size_size_list_nat @ ( concat_nat @ Xs ) ) ) ).

% length_concat_rev
thf(fact_3763_length__concat__rev,axiom,
    ! [Xs: list_list_int] :
      ( ( size_size_list_int @ ( concat_int @ ( rev_list_int @ Xs ) ) )
      = ( size_size_list_int @ ( concat_int @ Xs ) ) ) ).

% length_concat_rev
thf(fact_3764_nth__append__length,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn,Ys: list_P8527749157015355191n_assn] :
      ( ( nth_Pr1769885009046257848n_assn @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ Ys ) ) @ ( size_s6829681357464350627n_assn @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_3765_nth__append__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_3766_nth__append__length,axiom,
    ! [Xs: list_int,X: int,Ys: list_int] :
      ( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_3767_Suc__diff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_diff
thf(fact_3768_nth__map,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn,F: produc6575502325842934193n_assn > assn] :
      ( ( ord_less_nat @ N @ ( size_s6829681357464350627n_assn @ Xs ) )
     => ( ( nth_assn @ ( map_Pr8991440229025900053n_assn @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr1769885009046257848n_assn @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_3769_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > int] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_int @ ( map_nat_int @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_3770_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_3771_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: int > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_nat @ ( map_int_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_int @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_3772_nth__map,axiom,
    ! [N: nat,Xs: list_int,F: int > int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( map_int_int @ F @ Xs ) @ N )
        = ( F @ ( nth_int @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_3773_nth__append__first,axiom,
    ! [I: nat,L: list_nat,L3: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ L ) )
     => ( ( nth_nat @ ( append_nat @ L @ L3 ) @ I )
        = ( nth_nat @ L @ I ) ) ) ).

% nth_append_first
thf(fact_3774_nth__append__first,axiom,
    ! [I: nat,L: list_int,L3: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ L ) )
     => ( ( nth_int @ ( append_int @ L @ L3 ) @ I )
        = ( nth_int @ L @ I ) ) ) ).

% nth_append_first
thf(fact_3775_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_3776_nth__append__length__plus,axiom,
    ! [Xs: list_int,Ys: list_int,N: nat] :
      ( ( nth_int @ ( append_int @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_int @ Xs ) @ N ) )
      = ( nth_int @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_3777_length__butlast,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( butlast_nat @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_3778_length__butlast,axiom,
    ! [Xs: list_int] :
      ( ( size_size_list_int @ ( butlast_int @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ one_one_nat ) ) ).

% length_butlast
thf(fact_3779_rotate1__length01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_3780_rotate1__length01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( ( rotate1_int @ Xs )
        = Xs ) ) ).

% rotate1_length01
thf(fact_3781_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_nat @ Xs ) )
       => ( ( size_size_list_nat @ ( slice_nat @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3782_slice__len,axiom,
    ! [From: nat,To: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ From @ To )
     => ( ( ord_less_eq_nat @ To @ ( size_size_list_int @ Xs ) )
       => ( ( size_size_list_int @ ( slice_int @ From @ To @ Xs ) )
          = ( minus_minus_nat @ To @ From ) ) ) ) ).

% slice_len
thf(fact_3783_length__ge__1__conv,axiom,
    ! [L: list_a] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ L ) )
      = ( L != nil_a ) ) ).

% length_ge_1_conv
thf(fact_3784_length__ge__1__conv,axiom,
    ! [L: list_b] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_b @ L ) )
      = ( L != nil_b ) ) ).

% length_ge_1_conv
thf(fact_3785_length__ge__1__conv,axiom,
    ! [L: list_P8527749157015355191n_assn] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_s6829681357464350627n_assn @ L ) )
      = ( L != nil_Pr5671120429643327159n_assn ) ) ).

% length_ge_1_conv
thf(fact_3786_length__ge__1__conv,axiom,
    ! [L: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_nat @ L ) )
      = ( L != nil_nat ) ) ).

% length_ge_1_conv
thf(fact_3787_length__ge__1__conv,axiom,
    ! [L: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_int @ L ) )
      = ( L != nil_int ) ) ).

% length_ge_1_conv
thf(fact_3788_length__Cons,axiom,
    ! [X: produc6575502325842934193n_assn,Xs: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn @ ( cons_P2971678138204555879n_assn @ X @ Xs ) )
      = ( suc @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ).

% length_Cons
thf(fact_3789_length__Cons,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_Cons
thf(fact_3790_length__Cons,axiom,
    ! [X: int,Xs: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_Cons
thf(fact_3791_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ( suc @ N )
        = ( size_s6829681357464350627n_assn @ Xs ) )
      = ( ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_3792_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs ) )
      = ( ? [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_3793_Suc__length__conv,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs ) )
      = ( ? [Y3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y3 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_3794_length__Suc__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ Y3 @ Ys3 ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_3795_length__Suc__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_3796_length__Suc__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ Y3 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_3797_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_3798_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_int] :
      ( ( size_size_list_int @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_3799_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
thf(fact_3800_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
thf(fact_3801_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3802_sorted__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ ( suc @ I2 ) ) ) ) ) ) ).

% sorted_iff_nth_Suc
thf(fact_3803_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_3804_sorted__wrt__nth__less,axiom,
    ! [P: nat > nat > $o,Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3805_sorted__wrt__nth__less,axiom,
    ! [P: int > int > $o,Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ P @ Xs )
     => ( ( ord_less_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_wrt_nth_less
thf(fact_3806_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_nat
    = ( ^ [P3: nat > nat > $o,Xs3: list_nat] :
        ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
           => ( P3 @ ( nth_nat @ Xs3 @ I2 ) @ ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3807_sorted__wrt__iff__nth__less,axiom,
    ( sorted_wrt_int
    = ( ^ [P3: int > int > $o,Xs3: list_int] :
        ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs3 ) )
           => ( P3 @ ( nth_int @ Xs3 @ I2 ) @ ( nth_int @ Xs3 @ J2 ) ) ) ) ) ) ).

% sorted_wrt_iff_nth_less
thf(fact_3808_sorted__wrt01,axiom,
    ! [Xs: list_nat,P: nat > nat > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3809_sorted__wrt01,axiom,
    ! [Xs: list_int,P: int > int > $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ P @ Xs ) ) ).

% sorted_wrt01
thf(fact_3810_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_P8527749157015355191n_assn] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6829681357464350627n_assn @ Xs ) )
      = ( ? [X3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( cons_P2971678138204555879n_assn @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6829681357464350627n_assn @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_3811_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
      = ( ? [X3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_3812_Suc__le__length__iff,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
      = ( ? [X3: int,Ys3: list_int] :
            ( ( Xs
              = ( cons_int @ X3 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_3813_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: a > a > $o] : ( sorted_wrt_a @ P @ nil_a ) ).

% sorted_wrt.simps(1)
thf(fact_3814_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: b > b > $o] : ( sorted_wrt_b @ P @ nil_b ) ).

% sorted_wrt.simps(1)
thf(fact_3815_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o] : ( sorted3986126766855596574n_assn @ P @ nil_Pr5671120429643327159n_assn ) ).

% sorted_wrt.simps(1)
thf(fact_3816_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: nat > nat > $o] : ( sorted_wrt_nat @ P @ nil_nat ) ).

% sorted_wrt.simps(1)
thf(fact_3817_sorted__wrt_Osimps_I1_J,axiom,
    ! [P: int > int > $o] : ( sorted_wrt_int @ P @ nil_int ) ).

% sorted_wrt.simps(1)
thf(fact_3818_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys6: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys6 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys6 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_3819_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys6: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys6 ) @ ( size_size_list_int @ Xs2 ) )
             => ( P @ Ys6 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_3820_finite__maxlen,axiom,
    ! [M: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M )
     => ? [N7: nat] :
        ! [X6: list_nat] :
          ( ( member_list_nat @ X6 @ M )
         => ( ord_less_nat @ ( size_size_list_nat @ X6 ) @ N7 ) ) ) ).

% finite_maxlen
thf(fact_3821_finite__maxlen,axiom,
    ! [M: set_list_int] :
      ( ( finite3922522038869484883st_int @ M )
     => ? [N7: nat] :
        ! [X6: list_int] :
          ( ( member_list_int @ X6 @ M )
         => ( ord_less_nat @ ( size_size_list_int @ X6 ) @ N7 ) ) ) ).

% finite_maxlen
thf(fact_3822_map__eq__imp__length__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G2: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G2 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3823_map__eq__imp__length__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G2: nat > assn,Ys: list_nat] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( map_nat_assn @ G2 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3824_map__eq__imp__length__eq,axiom,
    ! [F: produc6575502325842934193n_assn > assn,Xs: list_P8527749157015355191n_assn,G2: int > assn,Ys: list_int] :
      ( ( ( map_Pr8991440229025900053n_assn @ F @ Xs )
        = ( map_int_assn @ G2 @ Ys ) )
     => ( ( size_s6829681357464350627n_assn @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3825_map__eq__imp__length__eq,axiom,
    ! [F: nat > assn,Xs: list_nat,G2: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_nat_assn @ F @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3826_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G2: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3827_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G2: int > nat,Ys: list_int] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_int_nat @ G2 @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3828_map__eq__imp__length__eq,axiom,
    ! [F: int > assn,Xs: list_int,G2: produc6575502325842934193n_assn > assn,Ys: list_P8527749157015355191n_assn] :
      ( ( ( map_int_assn @ F @ Xs )
        = ( map_Pr8991440229025900053n_assn @ G2 @ Ys ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_s6829681357464350627n_assn @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3829_map__eq__imp__length__eq,axiom,
    ! [F: int > nat,Xs: list_int,G2: nat > nat,Ys: list_nat] :
      ( ( ( map_int_nat @ F @ Xs )
        = ( map_nat_nat @ G2 @ Ys ) )
     => ( ( size_size_list_int @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_3830_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( rev_nat @ Xs ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_nat @ Xs ) )
           => ( ord_less_eq_nat @ ( nth_nat @ Xs @ ( suc @ I2 ) ) @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_3831_sorted__rev__iff__nth__Suc,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( rev_int @ Xs ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ ( suc @ I2 ) @ ( size_size_list_int @ Xs ) )
           => ( ord_less_eq_int @ ( nth_int @ Xs @ ( suc @ I2 ) ) @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).

% sorted_rev_iff_nth_Suc
thf(fact_3832_sorted__wrt__mergesort__by__rel,axiom,
    ! [R: nat > nat > $o,Xs: list_nat] :
      ( ! [X2: nat,Y2: nat] :
          ( ( R @ X2 @ Y2 )
          | ( R @ Y2 @ X2 ) )
     => ( ! [X2: nat,Y2: nat,Z3: nat] :
            ( ( R @ X2 @ Y2 )
           => ( ( R @ Y2 @ Z3 )
             => ( R @ X2 @ Z3 ) ) )
       => ( sorted_wrt_nat @ R @ ( mergesort_by_rel_nat @ R @ Xs ) ) ) ) ).

% sorted_wrt_mergesort_by_rel
thf(fact_3833_sorted__wrt__mergesort__by__rel,axiom,
    ! [R: int > int > $o,Xs: list_int] :
      ( ! [X2: int,Y2: int] :
          ( ( R @ X2 @ Y2 )
          | ( R @ Y2 @ X2 ) )
     => ( ! [X2: int,Y2: int,Z3: int] :
            ( ( R @ X2 @ Y2 )
           => ( ( R @ Y2 @ Z3 )
             => ( R @ X2 @ Z3 ) ) )
       => ( sorted_wrt_int @ R @ ( mergesort_by_rel_int @ R @ Xs ) ) ) ) ).

% sorted_wrt_mergesort_by_rel
thf(fact_3834_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I2: nat,J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3835_sorted__iff__nth__mono__less,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I2: nat,J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono_less
thf(fact_3836_sorted01,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% sorted01
thf(fact_3837_sorted01,axiom,
    ! [Xs: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ one_one_nat )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% sorted01
thf(fact_3838_list_Osize_I4_J,axiom,
    ! [X21: produc6575502325842934193n_assn,X22: list_P8527749157015355191n_assn] :
      ( ( size_s6829681357464350627n_assn @ ( cons_P2971678138204555879n_assn @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s6829681357464350627n_assn @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_3839_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_3840_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_3841_length__Suc__rev__conv,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys3: list_a,Y3: a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
            & ( ( size_size_list_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_3842_length__Suc__rev__conv,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys3: list_b,Y3: b] :
            ( ( Xs
              = ( append_b @ Ys3 @ ( cons_b @ Y3 @ nil_b ) ) )
            & ( ( size_size_list_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_3843_length__Suc__rev__conv,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys3: list_P8527749157015355191n_assn,Y3: produc6575502325842934193n_assn] :
            ( ( Xs
              = ( append282499809098378956n_assn @ Ys3 @ ( cons_P2971678138204555879n_assn @ Y3 @ nil_Pr5671120429643327159n_assn ) ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_3844_length__Suc__rev__conv,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys3: list_nat,Y3: nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_3845_length__Suc__rev__conv,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Ys3: list_int,Y3: int] :
            ( ( Xs
              = ( append_int @ Ys3 @ ( cons_int @ Y3 @ nil_int ) ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_rev_conv
thf(fact_3846_length__Suc__conv__rev,axiom,
    ! [Xs: list_a,N: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: a,Ys3: list_a] :
            ( ( Xs
              = ( append_a @ Ys3 @ ( cons_a @ Y3 @ nil_a ) ) )
            & ( ( size_size_list_a @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_3847_length__Suc__conv__rev,axiom,
    ! [Xs: list_b,N: nat] :
      ( ( ( size_size_list_b @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: b,Ys3: list_b] :
            ( ( Xs
              = ( append_b @ Ys3 @ ( cons_b @ Y3 @ nil_b ) ) )
            & ( ( size_size_list_b @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_3848_length__Suc__conv__rev,axiom,
    ! [Xs: list_P8527749157015355191n_assn,N: nat] :
      ( ( ( size_s6829681357464350627n_assn @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: produc6575502325842934193n_assn,Ys3: list_P8527749157015355191n_assn] :
            ( ( Xs
              = ( append282499809098378956n_assn @ Ys3 @ ( cons_P2971678138204555879n_assn @ Y3 @ nil_Pr5671120429643327159n_assn ) ) )
            & ( ( size_s6829681357464350627n_assn @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_3849_length__Suc__conv__rev,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ Y3 @ nil_nat ) ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_3850_length__Suc__conv__rev,axiom,
    ! [Xs: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs )
        = ( suc @ N ) )
      = ( ? [Y3: int,Ys3: list_int] :
            ( ( Xs
              = ( append_int @ Ys3 @ ( cons_int @ Y3 @ nil_int ) ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_3851_length__append__singleton,axiom,
    ! [Xs: list_a,X: a] :
      ( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
      = ( suc @ ( size_size_list_a @ Xs ) ) ) ).

% length_append_singleton
thf(fact_3852_length__append__singleton,axiom,
    ! [Xs: list_b,X: b] :
      ( ( size_size_list_b @ ( append_b @ Xs @ ( cons_b @ X @ nil_b ) ) )
      = ( suc @ ( size_size_list_b @ Xs ) ) ) ).

% length_append_singleton
thf(fact_3853_length__append__singleton,axiom,
    ! [Xs: list_P8527749157015355191n_assn,X: produc6575502325842934193n_assn] :
      ( ( size_s6829681357464350627n_assn @ ( append282499809098378956n_assn @ Xs @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) )
      = ( suc @ ( size_s6829681357464350627n_assn @ Xs ) ) ) ).

% length_append_singleton
thf(fact_3854_length__append__singleton,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) ) )
      = ( suc @ ( size_size_list_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_3855_length__append__singleton,axiom,
    ! [Xs: list_int,X: int] :
      ( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
      = ( suc @ ( size_size_list_int @ Xs ) ) ) ).

% length_append_singleton
thf(fact_3856_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3857_strict__sorted__imp__sorted,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ Xs ) ) ).

% strict_sorted_imp_sorted
thf(fact_3858_sorted__nth__mono,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3859_sorted__nth__mono,axiom,
    ! [Xs: list_int,I: nat,J: nat] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
         => ( ord_less_eq_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Xs @ J ) ) ) ) ) ).

% sorted_nth_mono
thf(fact_3860_sorted__iff__nth__mono,axiom,
    ! [Xs: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
      = ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
             => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3861_sorted__iff__nth__mono,axiom,
    ! [Xs: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
      = ( ! [I2: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I2 @ J2 )
           => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs ) )
             => ( ord_less_eq_int @ ( nth_int @ Xs @ I2 ) @ ( nth_int @ Xs @ J2 ) ) ) ) ) ) ).

% sorted_iff_nth_mono
thf(fact_3862_sorted2,axiom,
    ! [X: nat,Y: nat,Zs3: list_nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ X @ ( cons_nat @ Y @ Zs3 ) ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( sorted_wrt_nat @ ord_less_eq_nat @ ( cons_nat @ Y @ Zs3 ) ) ) ) ).

% sorted2
thf(fact_3863_sorted2,axiom,
    ! [X: int,Y: int,Zs3: list_int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ X @ ( cons_int @ Y @ Zs3 ) ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( sorted_wrt_int @ ord_less_eq_int @ ( cons_int @ Y @ Zs3 ) ) ) ) ).

% sorted2
thf(fact_3864_sorted0,axiom,
    sorted_wrt_nat @ ord_less_eq_nat @ nil_nat ).

% sorted0
thf(fact_3865_sorted0,axiom,
    sorted_wrt_int @ ord_less_eq_int @ nil_int ).

% sorted0
thf(fact_3866_strict__sorted__simps_I1_J,axiom,
    sorted_wrt_nat @ ord_less_nat @ nil_nat ).

% strict_sorted_simps(1)
thf(fact_3867_strict__sorted__simps_I1_J,axiom,
    sorted_wrt_int @ ord_less_int @ nil_int ).

% strict_sorted_simps(1)
thf(fact_3868_sorted__wrt1,axiom,
    ! [P: a > a > $o,X: a] : ( sorted_wrt_a @ P @ ( cons_a @ X @ nil_a ) ) ).

% sorted_wrt1
thf(fact_3869_sorted__wrt1,axiom,
    ! [P: b > b > $o,X: b] : ( sorted_wrt_b @ P @ ( cons_b @ X @ nil_b ) ) ).

% sorted_wrt1
thf(fact_3870_sorted__wrt1,axiom,
    ! [P: produc6575502325842934193n_assn > produc6575502325842934193n_assn > $o,X: produc6575502325842934193n_assn] : ( sorted3986126766855596574n_assn @ P @ ( cons_P2971678138204555879n_assn @ X @ nil_Pr5671120429643327159n_assn ) ) ).

% sorted_wrt1
thf(fact_3871_sorted__wrt1,axiom,
    ! [P: nat > nat > $o,X: nat] : ( sorted_wrt_nat @ P @ ( cons_nat @ X @ nil_nat ) ) ).

% sorted_wrt1
thf(fact_3872_sorted__wrt1,axiom,
    ! [P: int > int > $o,X: int] : ( sorted_wrt_int @ P @ ( cons_int @ X @ nil_int ) ) ).

% sorted_wrt1
thf(fact_3873_sorted__remove1,axiom,
    ! [Xs: list_nat,A: nat] :
      ( ( sorted_wrt_nat @ ord_less_eq_nat @ Xs )
     => ( sorted_wrt_nat @ ord_less_eq_nat @ ( remove1_nat @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_3874_sorted__remove1,axiom,
    ! [Xs: list_int,A: int] :
      ( ( sorted_wrt_int @ ord_less_eq_int @ Xs )
     => ( sorted_wrt_int @ ord_less_eq_int @ ( remove1_int @ A @ Xs ) ) ) ).

% sorted_remove1
thf(fact_3875_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( linord2612477271533052124et_int @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_3876_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_3877_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_int] : ( sorted_wrt_int @ ord_less_int @ ( linord2612477271533052124et_int @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_3878_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
    ! [A3: set_nat] : ( sorted_wrt_nat @ ord_less_nat @ ( linord2614967742042102400et_nat @ A3 ) ) ).

% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_3879_rev__nth,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( rev_nat @ Xs ) @ N )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3880_rev__nth,axiom,
    ! [N: nat,Xs: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( rev_int @ Xs ) @ N )
        = ( nth_int @ Xs @ ( minus_minus_nat @ ( size_size_list_int @ Xs ) @ ( suc @ N ) ) ) ) ) ).

% rev_nth
thf(fact_3881_sorted__mergesort__by__rel,axiom,
    ! [Xs: list_nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( mergesort_by_rel_nat @ ord_less_eq_nat @ Xs ) ) ).

% sorted_mergesort_by_rel
thf(fact_3882_sorted__mergesort__by__rel,axiom,
    ! [Xs: list_int] : ( sorted_wrt_int @ ord_less_eq_int @ ( mergesort_by_rel_int @ ord_less_eq_int @ Xs ) ) ).

% sorted_mergesort_by_rel
thf(fact_3883_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_a,P: list_nat > list_int > list_a > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_a @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_a @ nil_a )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: a,Ws3: list_a] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_a @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3884_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_b,P: list_nat > list_int > list_a > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_b @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_a @ nil_b )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: b,Ws3: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_b @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_b @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3885_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_a,P: list_nat > list_int > list_b > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_a @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_b @ nil_a )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: a,Ws3: list_a] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_a @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_a @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3886_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_b,P: list_nat > list_int > list_b > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_b @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_b @ nil_b )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: b,Ws3: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_b @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_b @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3887_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_nat,P: list_nat > list_int > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_a @ nil_nat )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_nat @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3888_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_nat,P: list_nat > list_int > list_b > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_nat @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_b @ nil_nat )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: nat,Ws3: list_nat] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_nat @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_nat @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3889_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_a,Ws2: list_int,P: list_nat > list_int > list_a > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_int @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_a @ nil_int )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: a,Zs2: list_a,W3: int,Ws3: list_int] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_int @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_int @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3890_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_b,Ws2: list_int,P: list_nat > list_int > list_b > list_int > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_b @ Zs3 ) )
       => ( ( ( size_size_list_b @ Zs3 )
            = ( size_size_list_int @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_b @ nil_int )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: b,Zs2: list_b,W3: int,Ws3: list_int] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_b @ Zs2 ) )
                   => ( ( ( size_size_list_b @ Zs2 )
                        = ( size_size_list_int @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_b @ Z3 @ Zs2 ) @ ( cons_int @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3891_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_nat,Ws2: list_a,P: list_nat > list_int > list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_a @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_nat @ nil_a )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: nat,Zs2: list_nat,W3: a,Ws3: list_a] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_a @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_a @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3892_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_int,Zs3: list_nat,Ws2: list_b,P: list_nat > list_int > list_nat > list_b > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( ( size_size_list_int @ Ys )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_b @ Ws2 ) )
         => ( ( P @ nil_nat @ nil_int @ nil_nat @ nil_b )
           => ( ! [X2: nat,Xs2: list_nat,Y2: int,Ys2: list_int,Z3: nat,Zs2: list_nat,W3: b,Ws3: list_b] :
                  ( ( ( size_size_list_nat @ Xs2 )
                    = ( size_size_list_int @ Ys2 ) )
                 => ( ( ( size_size_list_int @ Ys2 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_b @ Ws3 ) )
                     => ( ( P @ Xs2 @ Ys2 @ Zs2 @ Ws3 )
                       => ( P @ ( cons_nat @ X2 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_b @ W3 @ Ws3 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs3 @ Ws2 ) ) ) ) ) ) ).

% list_induct4
thf(fact_3893_nat__compl__induct_H,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N7: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N7 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N7 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct'
thf(fact_3894_nat__compl__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N7: nat] :
            ( ! [Nn: nat] :
                ( ( ord_less_eq_nat @ Nn @ N7 )
               => ( P @ Nn ) )
           => ( P @ ( suc @ N7 ) ) )
       => ( P @ N ) ) ) ).

% nat_compl_induct
thf(fact_3895_nat__in__between__eq_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_nat @ A @ B )
        & ( ord_less_eq_nat @ B @ ( suc @ A ) ) )
      = ( B
        = ( suc @ A ) ) ) ).

% nat_in_between_eq(1)
thf(fact_3896_nat__in__between__eq_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_nat @ B @ ( suc @ A ) ) )
      = ( B = A ) ) ).

% nat_in_between_eq(2)
thf(fact_3897_Suc__to__right,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( suc @ N )
        = M2 )
     => ( N
        = ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ).

% Suc_to_right
thf(fact_3898_nz__le__conv__less,axiom,
    ! [K: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ K @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ K @ ( suc @ zero_zero_nat ) ) @ M2 ) ) ) ).

% nz_le_conv_less
thf(fact_3899_Suc__n__minus__m__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ one_one_nat @ M2 )
       => ( ( suc @ ( minus_minus_nat @ N @ M2 ) )
          = ( minus_minus_nat @ N @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ).

% Suc_n_minus_m_eq
thf(fact_3900_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_3901_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_3902_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J: nat,I: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_3903_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_3904_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
        = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_3905_minus__assn__def,axiom,
    ( minus_minus_assn
    = ( ^ [A2: assn,B2: assn] : ( inf_inf_assn @ A2 @ ( uminus_uminus_assn @ B2 ) ) ) ) ).

% minus_assn_def
thf(fact_3906_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_3907_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_3908_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_3909_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_3910_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_3911_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_3912_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_3913_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( ( upt @ zero_zero_nat @ J )
        = nil_nat )
      = ( J = zero_zero_nat ) ) ).

% upt_0_eq_Nil_conv
thf(fact_3914_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_3915_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ J @ K ) )
     => ( ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
        = ( upt @ I @ K ) ) ) ).

% upt_merge
thf(fact_3916_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_3917_last__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( last_nat @ ( upt @ I @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_3918_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_3919_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_3920_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_3921_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_3922_butlast__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( butlast_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% butlast_upt
thf(fact_3923_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N3 ) ) ) ) ).

% atLeast_upt
thf(fact_3924_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_3925_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_3926_sorted__wrt__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M2 @ N ) ) ).

% sorted_wrt_upt
thf(fact_3927_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_3928_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M3 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_3929_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( upt @ I @ J )
          = ( append_nat @ Xs @ Ys ) )
        = ( ? [K4: nat] :
              ( ( ord_less_eq_nat @ I @ K4 )
              & ( ord_less_eq_nat @ K4 @ J )
              & ( ( upt @ I @ K4 )
                = Xs )
              & ( ( upt @ K4 @ J )
                = Ys ) ) ) ) ) ).

% upt_eq_append_conv
thf(fact_3930_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M3 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_3931_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_3932_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_3933_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( append_nat @ ( upt @ zero_zero_nat @ I ) @ ( upt @ I @ J ) )
        = ( upt @ zero_zero_nat @ J ) ) ) ).

% upt_append
thf(fact_3934_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_3935_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_3936_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_3937_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list_nat,I: nat,Is2: list_nat] :
      ( ( ( upt @ L @ H )
        = ( append_nat @ Is1 @ ( cons_nat @ I @ Is2 ) ) )
      = ( ( Is1
          = ( upt @ L @ I ) )
        & ( Is2
          = ( upt @ ( suc @ I ) @ H ) )
        & ( ord_less_eq_nat @ L @ I )
        & ( ord_less_nat @ I @ H ) ) ) ).

% upt_eq_lel_conv
thf(fact_3938_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_3939_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_3940_sorted__list__of__set__range,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sorted_list_of_set_range
thf(fact_3941_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I2: nat,J2: nat] : ( set_nat2 @ ( upt @ I2 @ J2 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_3942_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_3943_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_3944_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J2: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J2 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) @ ( cons_int @ J2 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_3945_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_3946_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_3947_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_3948_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_3949_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I @ J ) @ K )
        = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_3950_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_3951_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_3952_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_3953_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_3954_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_3955_sorted__upto,axiom,
    ! [M2: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M2 @ N ) ) ).

% sorted_upto
thf(fact_3956_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_3957_upto__code,axiom,
    ( upto
    = ( ^ [I2: int,J2: int] : ( upto_aux @ I2 @ J2 @ nil_int ) ) ) ).

% upto_code
thf(fact_3958_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I2: int,J2: int] : ( append_int @ ( upto @ I2 @ J2 ) ) ) ) ).

% upto_aux_def
thf(fact_3959_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I2: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_3960_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I2: int,J2: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_3961_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_3962_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_3963_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I2: int,J2: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J2 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_3964_upto_Osimps,axiom,
    ( upto
    = ( ^ [I2: int,J2: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J2 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_3965_upto_Oelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X @ Xa )
         => ( Y
            = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_3966_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_3967_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( upto @ I @ J )
        = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_3968_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I @ K )
          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_3969_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
           => ( P @ M3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less
thf(fact_3970_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M3: nat] :
            ( ( ord_less_eq_nat @ M3 @ N )
            & ( P @ M3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat2 @ X3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less
thf(fact_3971_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M3: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M3 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_3972_map__add__upt_H,axiom,
    ! [Ofs: nat,A: nat,B: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ Ofs )
        @ ( upt @ A @ B ) )
      = ( upt @ ( plus_plus_nat @ A @ Ofs ) @ ( plus_plus_nat @ B @ Ofs ) ) ) ).

% map_add_upt'
thf(fact_3973_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_3974_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_3975_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J2: int] : ( set_int2 @ ( upto @ I2 @ J2 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_3976_upt__filter__extend,axiom,
    ! [U: nat,U2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ U @ U2 )
     => ( ! [I3: nat] :
            ( ( ( ord_less_eq_nat @ U @ I3 )
              & ( ord_less_nat @ I3 @ U2 ) )
           => ~ ( P @ I3 ) )
       => ( ( filter_nat @ P @ ( upt @ zero_zero_nat @ U ) )
          = ( filter_nat @ P @ ( upt @ zero_zero_nat @ U2 ) ) ) ) ) ).

% upt_filter_extend
thf(fact_3977_remdups__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( remdups_nat @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% remdups_upt
thf(fact_3978_sort__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X3: nat] : X3
        @ ( upt @ M2 @ N ) )
      = ( upt @ M2 @ N ) ) ).

% sort_upt
thf(fact_3979_sort__upto,axiom,
    ! [I: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X3: int] : X3
        @ ( upto @ I @ J ) )
      = ( upto @ I @ J ) ) ).

% sort_upto
thf(fact_3980_merge__true__star,axiom,
    ( ( times_times_assn @ top_top_assn @ top_top_assn )
    = top_top_assn ) ).

% merge_true_star
thf(fact_3981_assn__basic__inequalities_I1_J,axiom,
    top_top_assn != one_one_assn ).

% assn_basic_inequalities(1)
thf(fact_3982_assn__basic__inequalities_I5_J,axiom,
    top_top_assn != bot_bot_assn ).

% assn_basic_inequalities(5)
thf(fact_3983_norm__assertion__simps_I12_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ X @ top_top_assn )
      = top_top_assn ) ).

% norm_assertion_simps(12)
thf(fact_3984_norm__assertion__simps_I11_J,axiom,
    ! [X: assn] :
      ( ( sup_sup_assn @ top_top_assn @ X )
      = top_top_assn ) ).

% norm_assertion_simps(11)
thf(fact_3985_norm__assertion__simps_I3_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ top_top_assn @ X )
      = X ) ).

% norm_assertion_simps(3)
thf(fact_3986_norm__assertion__simps_I4_J,axiom,
    ! [X: assn] :
      ( ( inf_inf_assn @ X @ top_top_assn )
      = X ) ).

% norm_assertion_simps(4)
thf(fact_3987_merge__true__star__ctx,axiom,
    ! [P: assn] :
      ( ( times_times_assn @ top_top_assn @ ( times_times_assn @ top_top_assn @ P ) )
      = ( times_times_assn @ top_top_assn @ P ) ) ).

% merge_true_star_ctx
thf(fact_3988_entails__solve__finalize_I1_J,axiom,
    ! [M: list_P8527749157015355191n_assn,P: assn] : ( fI_RESULT @ M @ P @ one_one_assn @ top_top_assn ) ).

% entails_solve_finalize(1)
thf(fact_3989_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_3990_upto_Opelims,axiom,
    ! [X: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X @ Xa )
               => ( Y
                  = ( cons_int @ X @ ( upto @ ( plus_plus_int @ X @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_3991_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [I3: int,J3: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J3 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) )
             => ( P @ I3 @ J3 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_3992_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
     => ( ( ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I @ J )
         => ( ( upto @ I @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_3993_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_3994_list__ex__iff__not__all__inverval__int,axiom,
    ! [P: int > $o,I: int,J: int] :
      ( ( list_ex_int @ P @ ( upto @ I @ J ) )
      = ( ~ ( all_interval_int @ ( comp_o_o_int @ (~) @ P ) @ I @ J ) ) ) ).

% list_ex_iff_not_all_inverval_int
thf(fact_3995_all__interval__int__def,axiom,
    ( all_interval_int
    = ( ^ [P3: int > $o,I2: int,J2: int] :
        ! [X3: int] :
          ( ( member_int2 @ X3 @ ( set_or1266510415728281911st_int @ I2 @ J2 ) )
         => ( P3 @ X3 ) ) ) ) ).

% all_interval_int_def
thf(fact_3996_list__all__iff__all__interval__int,axiom,
    ! [P: int > $o,I: int,J: int] :
      ( ( list_all_int @ P @ ( upto @ I @ J ) )
      = ( all_interval_int @ P @ I @ J ) ) ).

% list_all_iff_all_interval_int
thf(fact_3997_list__all__iff__all__interval__nat,axiom,
    ! [P: nat > $o,I: nat,J: nat] :
      ( ( list_all_nat @ P @ ( upt @ I @ J ) )
      = ( all_interval_nat @ P @ I @ J ) ) ).

% list_all_iff_all_interval_nat
thf(fact_3998_all__interval__nat__def,axiom,
    ( all_interval_nat
    = ( ^ [P3: nat > $o,I2: nat,J2: nat] :
        ! [X3: nat] :
          ( ( member_nat2 @ X3 @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) )
         => ( P3 @ X3 ) ) ) ) ).

% all_interval_nat_def
thf(fact_3999_list__ex__iff__not__all__inverval__nat,axiom,
    ! [P: nat > $o,I: nat,J: nat] :
      ( ( list_ex_nat @ P @ ( upt @ I @ J ) )
      = ( ~ ( all_interval_nat @ ( comp_o_o_nat @ (~) @ P ) @ I @ J ) ) ) ).

% list_ex_iff_not_all_inverval_nat
thf(fact_4000_FI__RESULT__def,axiom,
    ( fI_RESULT
    = ( ^ [M4: list_P8527749157015355191n_assn,UP: assn,UQ: assn,F5: assn] :
          ( ! [X3: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ M4 ) )
             => ( produc7274209992780475162assn_o @ entails @ X3 ) )
         => ( entails @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M4 ) @ one_one_assn ) @ UP ) @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M4 ) @ one_one_assn ) @ UQ ) @ F5 ) ) ) ) ) ).

% FI_RESULT_def
thf(fact_4001_ent__pure__pre__iff,axiom,
    ! [P: assn,B: $o,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ Q )
      = ( B
       => ( entails @ P @ Q ) ) ) ).

% ent_pure_pre_iff
thf(fact_4002_ent__pure__pre__iff__sng,axiom,
    ! [B: $o,Q: assn] :
      ( ( entails @ ( pure_assn @ B ) @ Q )
      = ( B
       => ( entails @ one_one_assn @ Q ) ) ) ).

% ent_pure_pre_iff_sng
thf(fact_4003_ent__conjI,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ A3 @ C3 )
       => ( entails @ A3 @ ( inf_inf_assn @ B3 @ C3 ) ) ) ) ).

% ent_conjI
thf(fact_4004_ent__conjE1,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C3 ) ) ).

% ent_conjE1
thf(fact_4005_ent__conjE2,axiom,
    ! [B3: assn,C3: assn,A3: assn] :
      ( ( entails @ B3 @ C3 )
     => ( entails @ ( inf_inf_assn @ A3 @ B3 ) @ C3 ) ) ).

% ent_conjE2
thf(fact_4006_ent__trans,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ Q )
     => ( ( entails @ Q @ R )
       => ( entails @ P @ R ) ) ) ).

% ent_trans
thf(fact_4007_ent__refl,axiom,
    ! [P: assn] : ( entails @ P @ P ) ).

% ent_refl
thf(fact_4008_ent__iffI,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( ( entails @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% ent_iffI
thf(fact_4009_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ Q ) ) ).

% is_entails
thf(fact_4010_ent__star__mono,axiom,
    ! [P: assn,P7: assn,Q: assn,Q4: assn] :
      ( ( entails @ P @ P7 )
     => ( ( entails @ Q @ Q4 )
       => ( entails @ ( times_times_assn @ P @ Q ) @ ( times_times_assn @ P7 @ Q4 ) ) ) ) ).

% ent_star_mono
thf(fact_4011_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps2: assn,F2: assn,Q: assn] :
      ( ( entails @ P @ R )
     => ( ( entails @ Ps2 @ ( times_times_assn @ P @ F2 ) )
       => ( ( entails @ ( times_times_assn @ R @ F2 ) @ Q )
         => ( entails @ Ps2 @ Q ) ) ) ) ).

% ent_frame_fwd
thf(fact_4012_fr__rot__rhs,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ B3 @ C3 ) )
     => ( entails @ A3 @ ( times_times_assn @ C3 @ B3 ) ) ) ).

% fr_rot_rhs
thf(fact_4013_fr__refl,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( entails @ ( times_times_assn @ A3 @ C3 ) @ ( times_times_assn @ B3 @ C3 ) ) ) ).

% fr_refl
thf(fact_4014_fr__rot,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ ( times_times_assn @ A3 @ B3 ) @ C3 )
     => ( entails @ ( times_times_assn @ B3 @ A3 ) @ C3 ) ) ).

% fr_rot
thf(fact_4015_ent__true,axiom,
    ! [P: assn] : ( entails @ P @ top_top_assn ) ).

% ent_true
thf(fact_4016_ent__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : ( entails @ B3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI2_direct
thf(fact_4017_ent__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : ( entails @ A3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% ent_disjI1_direct
thf(fact_4018_ent__disjI2_H,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% ent_disjI2'
thf(fact_4019_ent__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entails @ A3 @ B3 )
     => ( entails @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% ent_disjI1'
thf(fact_4020_ent__disjI2,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ Q @ R ) ) ).

% ent_disjI2
thf(fact_4021_ent__disjI1,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ ( sup_sup_assn @ P @ Q ) @ R )
     => ( entails @ P @ R ) ) ).

% ent_disjI1
thf(fact_4022_ent__disjE,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entails @ A3 @ C3 )
     => ( ( entails @ B3 @ C3 )
       => ( entails @ ( sup_sup_assn @ A3 @ B3 ) @ C3 ) ) ) ).

% ent_disjE
thf(fact_4023_ent__false,axiom,
    ! [P: assn] : ( entails @ bot_bot_assn @ P ) ).

% ent_false
thf(fact_4024_ent__star__mono__true,axiom,
    ! [A3: assn,A10: assn,B3: assn,B9: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ A10 @ top_top_assn ) )
     => ( ( entails @ B3 @ ( times_times_assn @ B9 @ top_top_assn ) )
       => ( entails @ ( times_times_assn @ ( times_times_assn @ A3 @ B3 ) @ top_top_assn ) @ ( times_times_assn @ ( times_times_assn @ A10 @ B9 ) @ top_top_assn ) ) ) ) ).

% ent_star_mono_true
thf(fact_4025_ent__refl__true,axiom,
    ! [A3: assn] : ( entails @ A3 @ ( times_times_assn @ A3 @ top_top_assn ) ) ).

% ent_refl_true
thf(fact_4026_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entails @ ( times_times_assn @ P @ R ) @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(1)
thf(fact_4027_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% ent_true_drop(2)
thf(fact_4028_fi__match__entails,axiom,
    ! [M2: list_P8527749157015355191n_assn] :
      ( ! [X2: produc6575502325842934193n_assn] :
          ( ( member7957490590177025114n_assn @ X2 @ ( set_Pr1139785259514867910n_assn @ M2 ) )
         => ( produc7274209992780475162assn_o @ entails @ X2 ) )
     => ( entails @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M2 ) @ one_one_assn ) @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M2 ) @ one_one_assn ) ) ) ).

% fi_match_entails
thf(fact_4029_FI__QUERY__def,axiom,
    ( fI_QUERY
    = ( ^ [P3: assn,Q2: assn,F5: assn] : ( entails @ P3 @ ( times_times_assn @ Q2 @ F5 ) ) ) ) ).

% FI_QUERY_def
thf(fact_4030_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F2: assn] :
      ( ( fI_QUERY @ P @ Q @ F2 )
     => ( entails @ P @ ( times_times_assn @ Q @ F2 ) ) ) ).

% frame_inference_init
thf(fact_4031_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ one_one_assn )
     => ( entails @ P @ Q ) ) ).

% entails_solve_init(2)
thf(fact_4032_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( ( fI_QUERY @ P @ Q @ top_top_assn )
     => ( entails @ P @ ( times_times_assn @ Q @ top_top_assn ) ) ) ).

% entails_solve_init(1)
thf(fact_4033_FI__match,axiom,
    ! [P4: assn,Q3: assn,M2: list_P8527749157015355191n_assn,Ps: assn,Up: assn,Qs: assn,Uq: assn,F: assn] :
      ( ( entails @ P4 @ Q3 )
     => ( ( fi @ ( cons_P2971678138204555879n_assn @ ( produc118845697133431529n_assn @ P4 @ Q3 ) @ M2 ) @ ( times_times_assn @ Ps @ Up ) @ ( times_times_assn @ Qs @ Uq ) @ sln @ sln @ F )
       => ( fi @ M2 @ ( times_times_assn @ Ps @ P4 ) @ ( times_times_assn @ Qs @ Q3 ) @ Up @ Uq @ F ) ) ) ).

% FI_match
thf(fact_4034_FI__def,axiom,
    ( fi
    = ( ^ [M3: list_P8527749157015355191n_assn,P6: assn,Q5: assn,Up2: assn,Uq2: assn,F3: assn] :
          ( ! [X3: produc6575502325842934193n_assn] :
              ( ( member7957490590177025114n_assn @ X3 @ ( set_Pr1139785259514867910n_assn @ M3 ) )
             => ( produc7274209992780475162assn_o @ entails @ X3 ) )
         => ( entails @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc9167289414957590229n_assn @ M3 ) @ one_one_assn ) @ P6 ) @ Up2 ) @ ( times_times_assn @ ( times_times_assn @ ( times_times_assn @ ( foldr_assn_assn @ times_times_assn @ ( map_Pr8991440229025900053n_assn @ produc2051961928117032727n_assn @ M3 ) @ one_one_assn ) @ Q5 ) @ Uq2 ) @ F3 ) ) ) ) ) ).

% FI_def
thf(fact_4035_ent__wand__frameI,axiom,
    ! [Q: assn,R: assn,F2: assn,S: assn,P: assn,X5: assn] :
      ( ( entails @ ( times_times_assn @ ( wand_assn @ Q @ R ) @ F2 ) @ S )
     => ( ( entails @ P @ ( times_times_assn @ F2 @ X5 ) )
       => ( ( entails @ ( times_times_assn @ Q @ X5 ) @ R )
         => ( entails @ P @ S ) ) ) ) ).

% ent_wand_frameI
thf(fact_4036_ent__wandI,axiom,
    ! [Q: assn,P: assn,R: assn] :
      ( ( entails @ ( times_times_assn @ Q @ P ) @ R )
     => ( entails @ P @ ( wand_assn @ Q @ R ) ) ) ).

% ent_wandI
thf(fact_4037_ent__mp,axiom,
    ! [P: assn,Q: assn] : ( entails @ ( times_times_assn @ P @ ( wand_assn @ P @ Q ) ) @ Q ) ).

% ent_mp
thf(fact_4038_Rep__assn__inject,axiom,
    ! [X: assn,Y: assn] :
      ( ( ( rep_assn @ X )
        = ( rep_assn @ Y ) )
      = ( X = Y ) ) ).

% Rep_assn_inject
thf(fact_4039_mod__or__dist,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ H )
      = ( ( rep_assn @ P @ H )
        | ( rep_assn @ Q @ H ) ) ) ).

% mod_or_dist
thf(fact_4040_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(5)
thf(fact_4041_mod__pure__star__dist,axiom,
    ! [P: assn,B: $o,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ ( pure_assn @ B ) ) @ H )
      = ( ( rep_assn @ P @ H )
        & B ) ) ).

% mod_pure_star_dist
thf(fact_4042_mod__h__bot__iff_I7_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( sup_sup_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        | ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(7)
thf(fact_4043_mod__h__bot__iff_I6_J,axiom,
    ! [P: assn,Q: assn,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
        & ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ) ) ).

% mod_h_bot_iff(6)
thf(fact_4044_mod__pure,axiom,
    ! [B: $o,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( pure_assn @ B ) @ H )
      = ( ( ( produc8586169260539613262et_nat @ H )
          = bot_bot_set_nat )
        & B ) ) ).

% mod_pure
thf(fact_4045_mod__h__bot__iff_I1_J,axiom,
    ! [B: $o,H: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ ( pure_assn @ B ) @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = B ) ).

% mod_h_bot_iff(1)
thf(fact_4046_ent__false__iff,axiom,
    ! [P: assn] :
      ( ( entails @ P @ bot_bot_assn )
      = ( ! [H2: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H2 ) ) ) ).

% ent_false_iff
thf(fact_4047_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B: $o] :
      ( ( entails @ P @ ( times_times_assn @ Q @ ( pure_assn @ B ) ) )
      = ( ! [H2: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H2 )
           => B )
        & ( entails @ P @ Q ) ) ) ).

% ent_pure_post_iff
thf(fact_4048_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B: $o] :
      ( ( entails @ P @ ( pure_assn @ B ) )
      = ( ! [H2: produc3658429121746597890et_nat] :
            ( ( rep_assn @ P @ H2 )
           => B )
        & ( entails @ P @ one_one_assn ) ) ) ).

% ent_pure_post_iff_sng
thf(fact_4049_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) )
      = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ bot_bot_set_nat ) ) ) ).

% mod_h_bot_indep
thf(fact_4050_mod__and__dist,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( inf_inf_assn @ P @ Q ) @ H )
      = ( ( rep_assn @ P @ H )
        & ( rep_assn @ Q @ H ) ) ) ).

% mod_and_dist
thf(fact_4051_mod__starD,axiom,
    ! [A3: assn,B3: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H )
     => ? [H1: produc3658429121746597890et_nat,H22: produc3658429121746597890et_nat] :
          ( ( rep_assn @ A3 @ H1 )
          & ( rep_assn @ B3 @ H22 ) ) ) ).

% mod_starD
thf(fact_4052_mod__starE,axiom,
    ! [A: assn,B: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A @ B ) @ H )
     => ~ ( ? [X_1: produc3658429121746597890et_nat] : ( rep_assn @ A @ X_1 )
         => ! [H_2: produc3658429121746597890et_nat] :
              ~ ( rep_assn @ B @ H_2 ) ) ) ).

% mod_starE
thf(fact_4053_mod__false,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ~ ( rep_assn @ bot_bot_assn @ H ) ).

% mod_false
thf(fact_4054_ent__fwd,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat,Q: assn] :
      ( ( rep_assn @ P @ H )
     => ( ( entails @ P @ Q )
       => ( rep_assn @ Q @ H ) ) ) ).

% ent_fwd
thf(fact_4055_entailsD,axiom,
    ! [P: assn,Q: assn,H: produc3658429121746597890et_nat] :
      ( ( entails @ P @ Q )
     => ( ( rep_assn @ P @ H )
       => ( rep_assn @ Q @ H ) ) ) ).

% entailsD
thf(fact_4056_entailsI,axiom,
    ! [P: assn,Q: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P @ H4 )
         => ( rep_assn @ Q @ H4 ) )
     => ( entails @ P @ Q ) ) ).

% entailsI
thf(fact_4057_entails__def,axiom,
    ( entails
    = ( ^ [P3: assn,Q2: assn] :
        ! [H2: produc3658429121746597890et_nat] :
          ( ( rep_assn @ P3 @ H2 )
         => ( rep_assn @ Q2 @ H2 ) ) ) ) ).

% entails_def
thf(fact_4058_mod__frame__fwd,axiom,
    ! [Ps2: assn,H: produc3658429121746597890et_nat,P: assn,R: assn,F2: assn] :
      ( ( rep_assn @ Ps2 @ H )
     => ( ( entails @ P @ R )
       => ( ( entails @ Ps2 @ ( times_times_assn @ P @ F2 ) )
         => ( rep_assn @ ( times_times_assn @ R @ F2 ) @ H ) ) ) ) ).

% mod_frame_fwd
thf(fact_4059_star__assnI,axiom,
    ! [P: assn,H: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,As4: set_nat] :
      ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
     => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H @ As4 ) )
       => ( ( ( inf_inf_set_nat @ As2 @ As4 )
            = bot_bot_set_nat )
         => ( rep_assn @ ( times_times_assn @ P @ Q ) @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As2 @ As4 ) ) ) ) ) ) ).

% star_assnI
thf(fact_4060_mod__star__conv,axiom,
    ! [A3: assn,B3: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ A3 @ B3 ) @ H )
      = ( ? [Hr: heap_e7401611519738050253t_unit,As1: set_nat,As22: set_nat] :
            ( ( H
              = ( produc7507926704131184380et_nat @ Hr @ ( sup_sup_set_nat @ As1 @ As22 ) ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( rep_assn @ A3 @ ( produc7507926704131184380et_nat @ Hr @ As1 ) )
            & ( rep_assn @ B3 @ ( produc7507926704131184380et_nat @ Hr @ As22 ) ) ) ) ) ).

% mod_star_conv
thf(fact_4061_mod__star__trueE_H,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ( ( ( produc1824681642469235216et_nat @ H5 )
              = ( produc1824681642469235216et_nat @ H ) )
           => ( ( ord_less_eq_set_nat @ ( produc8586169260539613262et_nat @ H5 ) @ ( produc8586169260539613262et_nat @ H ) )
             => ~ ( rep_assn @ P @ H5 ) ) ) ) ).

% mod_star_trueE'
thf(fact_4062_mod__star__trueI,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H )
     => ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H ) ) ).

% mod_star_trueI
thf(fact_4063_mod__star__trueE,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( times_times_assn @ P @ top_top_assn ) @ H )
     => ~ ! [H5: produc3658429121746597890et_nat] :
            ~ ( rep_assn @ P @ H5 ) ) ).

% mod_star_trueE
thf(fact_4064_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( rep_assn @ top_top_assn @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% mod_h_bot_iff(2)
thf(fact_4065_mod__emp,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ one_one_assn @ H )
      = ( ( produc8586169260539613262et_nat @ H )
        = bot_bot_set_nat ) ) ).

% mod_emp
thf(fact_4066_mod__emp__simp,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( rep_assn @ one_one_assn @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% mod_emp_simp
thf(fact_4067_mod__not__dist,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ ( uminus_uminus_assn @ P ) @ H )
      = ( ( in_range @ H )
        & ~ ( rep_assn @ P @ H ) ) ) ).

% mod_not_dist
thf(fact_4068_in__range__empty,axiom,
    ! [H: heap_e7401611519738050253t_unit] : ( in_range @ ( produc7507926704131184380et_nat @ H @ bot_bot_set_nat ) ) ).

% in_range_empty
thf(fact_4069_mod__true,axiom,
    ! [H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ top_top_assn @ H )
      = ( in_range @ H ) ) ).

% mod_true
thf(fact_4070_in__range__dist__union,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat,As4: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ ( sup_sup_set_nat @ As2 @ As4 ) ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
        & ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) ) ) ) ).

% in_range_dist_union
thf(fact_4071_models__in__range,axiom,
    ! [P: assn,H: produc3658429121746597890et_nat] :
      ( ( rep_assn @ P @ H )
     => ( in_range @ H ) ) ).

% models_in_range
thf(fact_4072_in__range__subset,axiom,
    ! [As2: set_nat,As4: set_nat,H: heap_e7401611519738050253t_unit] :
      ( ( ord_less_eq_set_nat @ As2 @ As4 )
     => ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As4 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).

% in_range_subset
thf(fact_4073_one__assn__raw_Ocases,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc7507926704131184380et_nat @ H4 @ As ) ) ).

% one_assn_raw.cases
thf(fact_4074_times__assn__raw_Ocases,axiom,
    ! [X: produc2732055786443039994et_nat] :
      ~ ! [P2: produc3658429121746597890et_nat > $o,Q6: produc3658429121746597890et_nat > $o,H4: heap_e7401611519738050253t_unit,As: set_nat] :
          ( X
         != ( produc2245416461498447860et_nat @ P2 @ ( produc5001842942810119800et_nat @ Q6 @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ).

% times_assn_raw.cases
thf(fact_4075_wand__assnI,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat,Q: assn,R: assn] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
     => ( ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
            ( ( ( inf_inf_set_nat @ As2 @ As5 )
              = bot_bot_set_nat )
           => ( ( relH @ As2 @ H @ H5 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As2 ) )
               => ( ( rep_assn @ Q @ ( produc7507926704131184380et_nat @ H5 @ As5 ) )
                 => ( rep_assn @ R @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As2 @ As5 ) ) ) ) ) ) )
       => ( rep_assn @ ( wand_assn @ Q @ R ) @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).

% wand_assnI
thf(fact_4076_times__assn__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( times_assn_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H @ As2 ) )
      = ( ? [As1: set_nat,As22: set_nat] :
            ( ( As2
              = ( sup_sup_set_nat @ As1 @ As22 ) )
            & ( ( inf_inf_set_nat @ As1 @ As22 )
              = bot_bot_set_nat )
            & ( P @ ( produc7507926704131184380et_nat @ H @ As1 ) )
            & ( Q @ ( produc7507926704131184380et_nat @ H @ As22 ) ) ) ) ) ).

% times_assn_raw.simps
thf(fact_4077_times__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( Y
              = ( ~ ? [As1: set_nat,As22: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As1 @ As22 ) )
                      & ( ( inf_inf_set_nat @ As1 @ As22 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H4 @ As1 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As22 ) ) ) ) ) ) ) ).

% times_assn_raw.elims(1)
thf(fact_4078_relH__dist__union,axiom,
    ! [As2: set_nat,As4: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ ( sup_sup_set_nat @ As2 @ As4 ) @ H @ H3 )
      = ( ( relH @ As2 @ H @ H3 )
        & ( relH @ As4 @ H @ H3 ) ) ) ).

% relH_dist_union
thf(fact_4079_relH__subset,axiom,
    ! [Bs: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( relH @ Bs @ H @ H3 )
     => ( ( ord_less_eq_set_nat @ As2 @ Bs )
       => ( relH @ As2 @ H @ H3 ) ) ) ).

% relH_subset
thf(fact_4080_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
thf(fact_4081_relH__sym,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H @ H3 )
     => ( relH @ As2 @ H3 @ H ) ) ).

% relH_sym
thf(fact_4082_mod__relH,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit,P: assn] :
      ( ( relH @ As2 @ H @ H3 )
     => ( ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
        = ( rep_assn @ P @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ) ).

% mod_relH
thf(fact_4083_relH__refl,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
     => ( relH @ As2 @ H @ H ) ) ).

% relH_refl
thf(fact_4084_relH__in__rangeI_I1_J,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H @ H3 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ).

% relH_in_rangeI(1)
thf(fact_4085_relH__in__rangeI_I2_J,axiom,
    ! [As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( relH @ As2 @ H @ H3 )
     => ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ).

% relH_in_rangeI(2)
thf(fact_4086_times__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ? [As12: set_nat,As23: set_nat] :
                ( ( As
                  = ( sup_sup_set_nat @ As12 @ As23 ) )
                & ( ( inf_inf_set_nat @ As12 @ As23 )
                  = bot_bot_set_nat )
                & ( X @ ( produc7507926704131184380et_nat @ H4 @ As12 ) )
                & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As23 ) ) ) ) ) ).

% times_assn_raw.elims(3)
thf(fact_4087_times__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ~ ? [As13: set_nat,As24: set_nat] :
                  ( ( As
                    = ( sup_sup_set_nat @ As13 @ As24 ) )
                  & ( ( inf_inf_set_nat @ As13 @ As24 )
                    = bot_bot_set_nat )
                  & ( X @ ( produc7507926704131184380et_nat @ H4 @ As13 ) )
                  & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As24 ) ) ) ) ) ).

% times_assn_raw.elims(2)
thf(fact_4088_times__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) )
               => ? [As12: set_nat,As23: set_nat] :
                    ( ( As
                      = ( sup_sup_set_nat @ As12 @ As23 ) )
                    & ( ( inf_inf_set_nat @ As12 @ As23 )
                      = bot_bot_set_nat )
                    & ( X @ ( produc7507926704131184380et_nat @ H4 @ As12 ) )
                    & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As23 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(3)
thf(fact_4089_times__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( times_assn_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) )
               => ~ ? [As13: set_nat,As24: set_nat] :
                      ( ( As
                        = ( sup_sup_set_nat @ As13 @ As24 ) )
                      & ( ( inf_inf_set_nat @ As13 @ As24 )
                        = bot_bot_set_nat )
                      & ( X @ ( produc7507926704131184380et_nat @ H4 @ As13 ) )
                      & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As24 ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(2)
thf(fact_4090_times__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( times_assn_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( Y
                  = ( ? [As1: set_nat,As22: set_nat] :
                        ( ( As
                          = ( sup_sup_set_nat @ As1 @ As22 ) )
                        & ( ( inf_inf_set_nat @ As1 @ As22 )
                          = bot_bot_set_nat )
                        & ( X @ ( produc7507926704131184380et_nat @ H4 @ As1 ) )
                        & ( Xa @ ( produc7507926704131184380et_nat @ H4 @ As22 ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ times_assn_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ) ) ).

% times_assn_raw.pelims(1)
thf(fact_4091_wand__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
              & ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
                  ( ( ( ( inf_inf_set_nat @ As @ As5 )
                      = bot_bot_set_nat )
                    & ( relH @ As @ H4 @ H5 )
                    & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
                    & ( X @ ( produc7507926704131184380et_nat @ H5 @ As5 ) ) )
                 => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ).

% wand_raw.elims(3)
thf(fact_4092_wand__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                & ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
                    ( ( ( ( inf_inf_set_nat @ As @ As6 )
                        = bot_bot_set_nat )
                      & ( relH @ As @ H4 @ H6 )
                      & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                      & ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
                   => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ).

% wand_raw.elims(2)
thf(fact_4093_wand__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( Xb
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( Y
              = ( ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H4 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) ) ) ) ) ).

% wand_raw.elims(1)
thf(fact_4094_wand__raw_Osimps,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( wand_raw @ P @ Q @ ( produc7507926704131184380et_nat @ H @ As2 ) )
      = ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
        & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
            ( ( ( ( inf_inf_set_nat @ As2 @ As7 )
                = bot_bot_set_nat )
              & ( relH @ As2 @ H @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As2 ) )
              & ( P @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
           => ( Q @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As2 @ As7 ) ) ) ) ) ) ).

% wand_raw.simps
thf(fact_4095_wand__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ~ ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) )
               => ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                  & ! [H5: heap_e7401611519738050253t_unit,As5: set_nat] :
                      ( ( ( ( inf_inf_set_nat @ As @ As5 )
                          = bot_bot_set_nat )
                        & ( relH @ As @ H4 @ H5 )
                        & ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
                        & ( X @ ( produc7507926704131184380et_nat @ H5 @ As5 ) ) )
                     => ( Xa @ ( produc7507926704131184380et_nat @ H5 @ ( sup_sup_set_nat @ As @ As5 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(3)
thf(fact_4096_wand__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat] :
      ( ( wand_raw @ X @ Xa @ Xb )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) )
               => ~ ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                    & ! [H6: heap_e7401611519738050253t_unit,As6: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As6 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H4 @ H6 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H6 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H6 @ As6 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H6 @ ( sup_sup_set_nat @ As @ As6 ) ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(2)
thf(fact_4097_wand__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Xa: produc3658429121746597890et_nat > $o,Xb: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( wand_raw @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ Xb ) ) )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( Xb
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( Y
                  = ( ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) )
                    & ! [H7: heap_e7401611519738050253t_unit,As7: set_nat] :
                        ( ( ( ( inf_inf_set_nat @ As @ As7 )
                            = bot_bot_set_nat )
                          & ( relH @ As @ H4 @ H7 )
                          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As ) )
                          & ( X @ ( produc7507926704131184380et_nat @ H7 @ As7 ) ) )
                       => ( Xa @ ( produc7507926704131184380et_nat @ H7 @ ( sup_sup_set_nat @ As @ As7 ) ) ) ) ) )
               => ~ ( accp_P1862375125659990705et_nat @ wand_raw_rel @ ( produc2245416461498447860et_nat @ X @ ( produc5001842942810119800et_nat @ Xa @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ) ) ).

% wand_raw.pelims(1)
thf(fact_4098_uminus__assn__def,axiom,
    ( uminus_uminus_assn
    = ( ^ [P3: assn] :
          ( abs_assn
          @ ^ [H2: produc3658429121746597890et_nat] :
              ( ( in_range @ H2 )
              & ~ ( rep_assn @ P3 @ H2 ) ) ) ) ) ).

% uminus_assn_def
thf(fact_4099_Rep__assn__inverse,axiom,
    ! [X: assn] :
      ( ( abs_assn @ ( rep_assn @ X ) )
      = X ) ).

% Rep_assn_inverse
thf(fact_4100_pure__assn__def,axiom,
    ( pure_assn
    = ( ^ [B2: $o] : ( abs_assn @ ( pure_a825153325127701367it_nat @ B2 ) ) ) ) ).

% pure_assn_def
thf(fact_4101_Abs__assn__eqI_I2_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( P @ H4 )
          = ( rep_assn @ Pr @ H4 ) )
     => ( Pr
        = ( abs_assn @ P ) ) ) ).

% Abs_assn_eqI(2)
thf(fact_4102_Abs__assn__eqI_I1_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Pr: assn] :
      ( ! [H4: produc3658429121746597890et_nat] :
          ( ( P @ H4 )
          = ( rep_assn @ Pr @ H4 ) )
     => ( ( abs_assn @ P )
        = Pr ) ) ).

% Abs_assn_eqI(1)
thf(fact_4103_bot__assn__def,axiom,
    ( bot_bot_assn
    = ( abs_assn
      @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ) ).

% bot_assn_def
thf(fact_4104_top__assn__def,axiom,
    ( top_top_assn
    = ( abs_assn @ in_range ) ) ).

% top_assn_def
thf(fact_4105_sup__assn__def,axiom,
    ( sup_sup_assn
    = ( ^ [P3: assn,Q2: assn] :
          ( abs_assn
          @ ^ [H2: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P3 @ H2 )
              | ( rep_assn @ Q2 @ H2 ) ) ) ) ) ).

% sup_assn_def
thf(fact_4106_inf__assn__def,axiom,
    ( inf_inf_assn
    = ( ^ [P3: assn,Q2: assn] :
          ( abs_assn
          @ ^ [H2: produc3658429121746597890et_nat] :
              ( ( rep_assn @ P3 @ H2 )
              & ( rep_assn @ Q2 @ H2 ) ) ) ) ) ).

% inf_assn_def
thf(fact_4107_times__assn__def,axiom,
    ( times_times_assn
    = ( ^ [P3: assn,Q2: assn] : ( abs_assn @ ( times_assn_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).

% times_assn_def
thf(fact_4108_wand__assn__def,axiom,
    ( wand_assn
    = ( ^ [P3: assn,Q2: assn] : ( abs_assn @ ( wand_raw @ ( rep_assn @ P3 ) @ ( rep_assn @ Q2 ) ) ) ) ) ).

% wand_assn_def
thf(fact_4109_one__assn__def,axiom,
    ( one_one_assn
    = ( abs_assn @ one_assn_raw ) ) ).

% one_assn_def
thf(fact_4110_one__assn__raw_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( one_assn_raw @ ( produc7507926704131184380et_nat @ H @ As2 ) )
      = ( As2 = bot_bot_set_nat ) ) ).

% one_assn_raw.simps
thf(fact_4111_one__assn__raw_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( Y
              = ( As != bot_bot_set_nat ) ) ) ) ).

% one_assn_raw.elims(1)
thf(fact_4112_one__assn__raw_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( As = bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(3)
thf(fact_4113_one__assn__raw_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( As != bot_bot_set_nat ) ) ) ).

% one_assn_raw.elims(2)
thf(fact_4114_properI,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ! [As: set_nat,H4: heap_e7401611519738050253t_unit] :
          ( ( P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
         => ( in_range @ ( produc7507926704131184380et_nat @ H4 @ As ) ) )
     => ( ! [As: set_nat,H4: heap_e7401611519738050253t_unit,H5: heap_e7401611519738050253t_unit] :
            ( ( P @ ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( ( relH @ As @ H4 @ H5 )
             => ( ( in_range @ ( produc7507926704131184380et_nat @ H5 @ As ) )
               => ( P @ ( produc7507926704131184380et_nat @ H5 @ As ) ) ) ) )
       => ( proper @ P ) ) ) ).

% properI
thf(fact_4115_properD2,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat,H3: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
       => ( ( relH @ As2 @ H @ H3 )
         => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As2 ) )
           => ( P @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ) ) ) ).

% properD2
thf(fact_4116_proper__def,axiom,
    ( proper
    = ( ^ [P3: produc3658429121746597890et_nat > $o] :
        ! [H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit,As3: set_nat] :
          ( ( ( P3 @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
           => ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) ) )
          & ( ( ( P3 @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
              & ( relH @ As3 @ H2 @ H7 )
              & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As3 ) ) )
           => ( P3 @ ( produc7507926704131184380et_nat @ H7 @ As3 ) ) ) ) ) ) ).

% proper_def
thf(fact_4117_bool__assn__proper_I4_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H2: produc3658429121746597890et_nat] :
              ( ( P @ H2 )
              & ( Q @ H2 ) ) ) ) ) ).

% bool_assn_proper(4)
thf(fact_4118_bool__assn__proper_I3_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper
          @ ^ [H2: produc3658429121746597890et_nat] :
              ( ( P @ H2 )
              | ( Q @ H2 ) ) ) ) ) ).

% bool_assn_proper(3)
thf(fact_4119_bool__assn__proper_I2_J,axiom,
    ( proper
    @ ^ [Uu2: produc3658429121746597890et_nat] : $false ) ).

% bool_assn_proper(2)
thf(fact_4120_bool__assn__proper_I1_J,axiom,
    proper @ in_range ).

% bool_assn_proper(1)
thf(fact_4121_times__assn__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( ( proper @ Q )
       => ( proper @ ( times_assn_raw @ P @ Q ) ) ) ) ).

% times_assn_proper
thf(fact_4122_wand__proper,axiom,
    ! [P: produc3658429121746597890et_nat > $o,Q: produc3658429121746597890et_nat > $o] : ( proper @ ( wand_raw @ P @ Q ) ) ).

% wand_proper
thf(fact_4123_one__assn__proper,axiom,
    proper @ one_assn_raw ).

% one_assn_proper
thf(fact_4124_bool__assn__proper_I5_J,axiom,
    ! [P: produc3658429121746597890et_nat > $o] :
      ( ( proper @ P )
     => ( proper
        @ ^ [H2: produc3658429121746597890et_nat] :
            ( ( in_range @ H2 )
            & ~ ( P @ H2 ) ) ) ) ).

% bool_assn_proper(5)
thf(fact_4125_pure__assn__proper,axiom,
    ! [B: $o] : ( proper @ ( pure_a825153325127701367it_nat @ B ) ) ).

% pure_assn_proper
thf(fact_4126_Abs__assn__inject,axiom,
    ! [X: produc3658429121746597890et_nat > $o,Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ X @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
       => ( ( ( abs_assn @ X )
            = ( abs_assn @ Y ) )
          = ( X = Y ) ) ) ) ).

% Abs_assn_inject
thf(fact_4127_Abs__assn__induct,axiom,
    ! [P: assn > $o,X: assn] :
      ( ! [Y2: produc3658429121746597890et_nat > $o] :
          ( ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) )
         => ( P @ ( abs_assn @ Y2 ) ) )
     => ( P @ X ) ) ).

% Abs_assn_induct
thf(fact_4128_Abs__assn__cases,axiom,
    ! [X: assn] :
      ~ ! [Y2: produc3658429121746597890et_nat > $o] :
          ( ( X
            = ( abs_assn @ Y2 ) )
         => ~ ( member6576561426505652726_nat_o @ Y2 @ ( collec939566748876313656_nat_o @ proper ) ) ) ).

% Abs_assn_cases
thf(fact_4129_Rep__assn__induct,axiom,
    ! [Y: produc3658429121746597890et_nat > $o,P: ( produc3658429121746597890et_nat > $o ) > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ! [X2: assn] : ( P @ ( rep_assn @ X2 ) )
       => ( P @ Y ) ) ) ).

% Rep_assn_induct
thf(fact_4130_Rep__assn__cases,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ~ ! [X2: assn] :
            ( Y
           != ( rep_assn @ X2 ) ) ) ).

% Rep_assn_cases
thf(fact_4131_Rep__assn,axiom,
    ! [X: assn] : ( member6576561426505652726_nat_o @ ( rep_assn @ X ) @ ( collec939566748876313656_nat_o @ proper ) ) ).

% Rep_assn
thf(fact_4132_properD1,axiom,
    ! [P: produc3658429121746597890et_nat > $o,H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( proper @ P )
     => ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
       => ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) ) ) ) ).

% properD1
thf(fact_4133_Abs__assn__inverse,axiom,
    ! [Y: produc3658429121746597890et_nat > $o] :
      ( ( member6576561426505652726_nat_o @ Y @ ( collec939566748876313656_nat_o @ proper ) )
     => ( ( rep_assn @ ( abs_assn @ Y ) )
        = Y ) ) ).

% Abs_assn_inverse
thf(fact_4134_proper__iff,axiom,
    ! [P: produc3658429121746597890et_nat > $o,As2: set_nat,H: heap_e7401611519738050253t_unit,H3: heap_e7401611519738050253t_unit] :
      ( ( proper @ P )
     => ( ( relH @ As2 @ H @ H3 )
       => ( ( in_range @ ( produc7507926704131184380et_nat @ H3 @ As2 ) )
         => ( ( P @ ( produc7507926704131184380et_nat @ H @ As2 ) )
            = ( P @ ( produc7507926704131184380et_nat @ H3 @ As2 ) ) ) ) ) ) ).

% proper_iff
thf(fact_4135_entt__def__true,axiom,
    ( entailst
    = ( ^ [P3: assn,Q2: assn] : ( entails @ ( times_times_assn @ P3 @ top_top_assn ) @ ( times_times_assn @ Q2 @ top_top_assn ) ) ) ) ).

% entt_def_true
thf(fact_4136_entailst__def,axiom,
    ( entailst
    = ( ^ [A5: assn,B4: assn] : ( entails @ A5 @ ( times_times_assn @ B4 @ top_top_assn ) ) ) ) ).

% entailst_def
thf(fact_4137_enttI__true,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ ( times_times_assn @ P @ top_top_assn ) @ ( times_times_assn @ Q @ top_top_assn ) )
     => ( entailst @ P @ Q ) ) ).

% enttI_true
thf(fact_4138_entt__refl,axiom,
    ! [A3: assn] : ( entailst @ A3 @ A3 ) ).

% entt_refl
thf(fact_4139_entt__true,axiom,
    ! [A3: assn] : ( entailst @ A3 @ top_top_assn ) ).

% entt_true
thf(fact_4140_entt__emp,axiom,
    ! [A3: assn] : ( entailst @ A3 @ one_one_assn ) ).

% entt_emp
thf(fact_4141_entt__star__true__simp_I2_J,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ ( times_times_assn @ A3 @ top_top_assn ) @ B3 )
      = ( entailst @ A3 @ B3 ) ) ).

% entt_star_true_simp(2)
thf(fact_4142_entt__star__true__simp_I1_J,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) )
      = ( entailst @ A3 @ B3 ) ) ).

% entt_star_true_simp(1)
thf(fact_4143_entt__disjI2__direct,axiom,
    ! [B3: assn,A3: assn] : ( entailst @ B3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% entt_disjI2_direct
thf(fact_4144_entt__disjI1__direct,axiom,
    ! [A3: assn,B3: assn] : ( entailst @ A3 @ ( sup_sup_assn @ A3 @ B3 ) ) ).

% entt_disjI1_direct
thf(fact_4145_entt__disjI2_H,axiom,
    ! [A3: assn,C3: assn,B3: assn] :
      ( ( entailst @ A3 @ C3 )
     => ( entailst @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% entt_disjI2'
thf(fact_4146_entt__disjI1_H,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( entailst @ A3 @ ( sup_sup_assn @ B3 @ C3 ) ) ) ).

% entt_disjI1'
thf(fact_4147_entt__disjD2,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ C3 )
     => ( entailst @ B3 @ C3 ) ) ).

% entt_disjD2
thf(fact_4148_entt__disjD1,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ C3 )
     => ( entailst @ A3 @ C3 ) ) ).

% entt_disjD1
thf(fact_4149_entt__disjE,axiom,
    ! [A3: assn,M: assn,B3: assn] :
      ( ( entailst @ A3 @ M )
     => ( ( entailst @ B3 @ M )
       => ( entailst @ ( sup_sup_assn @ A3 @ B3 ) @ M ) ) ) ).

% entt_disjE
thf(fact_4150_entt__frame__fwd,axiom,
    ! [P: assn,Q: assn,A3: assn,F2: assn,B3: assn] :
      ( ( entailst @ P @ Q )
     => ( ( entailst @ A3 @ ( times_times_assn @ P @ F2 ) )
       => ( ( entailst @ ( times_times_assn @ Q @ F2 ) @ B3 )
         => ( entailst @ A3 @ B3 ) ) ) ) ).

% entt_frame_fwd
thf(fact_4151_entt__star__mono,axiom,
    ! [A3: assn,B3: assn,C3: assn,D: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( ( entailst @ C3 @ D )
       => ( entailst @ ( times_times_assn @ A3 @ C3 ) @ ( times_times_assn @ B3 @ D ) ) ) ) ).

% entt_star_mono
thf(fact_4152_entt__fr__refl,axiom,
    ! [F2: assn,F6: assn,A3: assn] :
      ( ( entailst @ F2 @ F6 )
     => ( entailst @ ( times_times_assn @ F2 @ A3 ) @ ( times_times_assn @ F6 @ A3 ) ) ) ).

% entt_fr_refl
thf(fact_4153_entt__fr__drop,axiom,
    ! [F2: assn,F6: assn,A3: assn] :
      ( ( entailst @ F2 @ F6 )
     => ( entailst @ ( times_times_assn @ F2 @ A3 ) @ F6 ) ) ).

% entt_fr_drop
thf(fact_4154_entt__trans,axiom,
    ! [A3: assn,B3: assn,C3: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( ( entailst @ B3 @ C3 )
       => ( entailst @ A3 @ C3 ) ) ) ).

% entt_trans
thf(fact_4155_ent__imp__entt,axiom,
    ! [P: assn,Q: assn] :
      ( ( entails @ P @ Q )
     => ( entailst @ P @ Q ) ) ).

% ent_imp_entt
thf(fact_4156_enttD,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entailst @ A3 @ B3 )
     => ( entails @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) ) ) ).

% enttD
thf(fact_4157_enttI,axiom,
    ! [A3: assn,B3: assn] :
      ( ( entails @ A3 @ ( times_times_assn @ B3 @ top_top_assn ) )
     => ( entailst @ A3 @ B3 ) ) ).

% enttI
thf(fact_4158_in__range_Oelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ! [X2: nat] :
                ( ( member_nat2 @ X2 @ As )
               => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H4 ) ) ) ) ) ).

% in_range.elims(3)
thf(fact_4159_in__range_Osimps,axiom,
    ! [H: heap_e7401611519738050253t_unit,As2: set_nat] :
      ( ( in_range @ ( produc7507926704131184380et_nat @ H @ As2 ) )
      = ( ! [X3: nat] :
            ( ( member_nat2 @ X3 @ As2 )
           => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H ) ) ) ) ) ).

% in_range.simps
thf(fact_4160_in__range_Oelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ( Y
              = ( ~ ! [X3: nat] :
                      ( ( member_nat2 @ X3 @ As )
                     => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.elims(1)
thf(fact_4161_in__range_Oelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
            ( ( X
              = ( produc7507926704131184380et_nat @ H4 @ As ) )
           => ~ ! [X6: nat] :
                  ( ( member_nat2 @ X6 @ As )
                 => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H4 ) ) ) ) ) ).

% in_range.elims(2)
thf(fact_4162_type__definition__assn,axiom,
    type_d3909072315231072503_nat_o @ rep_assn @ abs_assn @ ( collec939566748876313656_nat_o @ proper ) ).

% type_definition_assn
thf(fact_4163_wait__rule,axiom,
    ! [N: nat] :
      ( hoare_8945653483474564448t_unit @ one_one_assn @ ( heap_Time_wait @ N )
      @ ^ [Uu2: product_unit] : one_one_assn ) ).

% wait_rule
thf(fact_4164_inf__unit__def,axiom,
    ( inf_inf_Product_unit
    = ( ^ [Uu3: product_unit,Uv: product_unit] : product_Unity ) ) ).

% inf_unit_def
thf(fact_4165_relH__def,axiom,
    ( relH
    = ( ^ [As3: set_nat,H2: heap_e7401611519738050253t_unit,H7: heap_e7401611519738050253t_unit] :
          ( ( in_range @ ( produc7507926704131184380et_nat @ H2 @ As3 ) )
          & ( in_range @ ( produc7507926704131184380et_nat @ H7 @ As3 ) )
          & ! [T2: typerep,X3: nat] :
              ( ( member_nat2 @ X3 @ As3 )
             => ( ( ( refs_Product_unit @ H2 @ T2 @ X3 )
                  = ( refs_Product_unit @ H7 @ T2 @ X3 ) )
                & ( ( arrays_Product_unit @ H2 @ T2 @ X3 )
                  = ( arrays_Product_unit @ H7 @ T2 @ X3 ) ) ) ) ) ) ) ).

% relH_def
thf(fact_4166_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) )
          @ ^ [Uu2: nat] :
              ( collect_nat
              @ ^ [X3: nat] : ( ord_less_nat @ X3 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_4167_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu2: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_4168_in__range_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( in_range @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( Y
                  = ( ! [X3: nat] :
                        ( ( member_nat2 @ X3 @ As )
                       => ( ord_less_nat @ X3 @ ( lim_Product_unit @ H4 ) ) ) ) )
               => ~ ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ).

% in_range.pelims(1)
thf(fact_4169_in__range_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ~ ! [X6: nat] :
                      ( ( member_nat2 @ X6 @ As )
                     => ( ord_less_nat @ X6 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.pelims(2)
thf(fact_4170_in__range_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( in_range @ X )
     => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ in_range_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ! [X2: nat] :
                    ( ( member_nat2 @ X2 @ As )
                   => ( ord_less_nat @ X2 @ ( lim_Product_unit @ H4 ) ) ) ) ) ) ) ).

% in_range.pelims(3)
thf(fact_4171_one__assn__raw_Opelims_I3_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ~ ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( As = bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(3)
thf(fact_4172_one__assn__raw_Opelims_I2_J,axiom,
    ! [X: produc3658429121746597890et_nat] :
      ( ( one_assn_raw @ X )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) )
               => ( As != bot_bot_set_nat ) ) ) ) ) ).

% one_assn_raw.pelims(2)
thf(fact_4173_one__assn__raw_Opelims_I1_J,axiom,
    ! [X: produc3658429121746597890et_nat,Y: $o] :
      ( ( ( one_assn_raw @ X )
        = Y )
     => ( ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ X )
       => ~ ! [H4: heap_e7401611519738050253t_unit,As: set_nat] :
              ( ( X
                = ( produc7507926704131184380et_nat @ H4 @ As ) )
             => ( ( Y
                  = ( As = bot_bot_set_nat ) )
               => ~ ( accp_P5801069581201407417et_nat @ one_assn_raw_rel @ ( produc7507926704131184380et_nat @ H4 @ As ) ) ) ) ) ) ).

% one_assn_raw.pelims(1)

% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Assertions__Oassn_T,axiom,
    ! [X: assn,Y: assn] :
      ( ( if_assn @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( vEBT_L4319891404334229444sn_a_b @ p @ nil_a @ l )
    = ( pure_assn @ ( l = nil_b ) ) ) ).

%------------------------------------------------------------------------------